# Are Error-Analysis Lessons Effective?

I recently came across a thought-provoking video where Simon Sinek explains that the human brain struggles to process negative statements. In the video, Sinek states that skiers should not spend their time thinking "don’t hit the trees" because all they’ll notice is the trees on the mountain. Instead, Sinek suggests that skiers should focus on the positive statement of "follow the path," which directs their attention to the desired outcome of navigating through the trees.

I was particularly intrigued by Sinek's idea because I am preparing to teach a lesson on error-analysis in solving trigonometric equations to my Pre-Calculus students. I have always found error-analysis lessons to be valuable in helping students build confidence and improve their performance on summative assessments. Additionally, a quick Google search reveals numerous benefits of incorporating error-analysis in mathematics education.

However, I feel that Sinek's idea raises a valid concern: since error-analysis lessons inherently involve showing students what not to do, could these types of lessons potentially hinder their mathematical development by triggering a focus on negative statements? Or is there a flaw in Sinek's argument? I am eager to hear your thoughts on this matter.

(Please note that my intention is not to discredit Sinek's idea, but rather to critically examine its potential implications in the context of mathematics education.)

• I do personally avoid "common errors" presentations in my own teaching (up until it happens multiple times in a classroom). I've heard this advice before, it seems to match my experience, but don't have research data on it. All it takes is one student seeing $(a + b)^2 = a^2 + b^2$ and misunderstanding the context ("this is an error") to be broken indefinitely. (It kind of came up in CS this week: a student with weak English skills saw me write 1/2 = 0 in integers on the board, and manually replaced the "1/2" in their physics code with 0's.) Commented Apr 23, 2023 at 17:08
• I recall reading, back as a volleyball coach, a study showing you should not train your hitters to hit at targets; rather, train them to hit between targets, which is what they should be doing during an actual match. It's unclear whether be-the-change-you-want-to-see can be generalized from sports to include math. Nonetheless, my opinion is that students finding and fixing mistakes (and justifying the fix) is better practice than showing them mistakes. -- I just thought of your question this morning since I have a few students who always find $\int x^2(3-x)\;dx={x^3\over3}(3x-{x^2\over2})+C$ Commented Apr 23, 2023 at 17:24
• @Raciquel I agree that the students need to fix the mistakes themselves as opposed to only seeing the common mistakes. I should have included above that my lesson requires the students to analyze sample solutions to trigonometric equations, identify the error(s), and then solve correctly. Commented Apr 23, 2023 at 18:32
• The question about math has a useful component. But don't put too much stock into Simon Sinek video... There's no reason to think that considering his flimsy analogies are worthwhile. He is very successful at selling himself though. forbes.com/sites/michaelschein/2018/06/13/… Commented Apr 25, 2023 at 12:13
• Sinek's idea, as presented here by you, is fundamentally flawed. In the example of the skier, the skier is taking the math test. In this case, error-analysis during a test (unless its on error-analysis) would not be helpful. However, the instructor will tell the skier of the hazards before the skier puts on their boots for the first time. And instructors will certainly look into a poor soul that smacked a tree to see if the course needs to be adjusted for future skiers. Instructors also teach techniques for avoiding trees or reducing the damage of an impact. Very flawed. Commented Apr 25, 2023 at 22:17

I prefer to do in the format "what's wrong with this argument/answer? Can you explain why it is invalid?" than in the format "here is a common mistake, lo and behold!", i.e., concentrating on the techniques of (self-) checking the solutions (from the trivial ones like "plug in the answer into the equation" and "check your identities on small integers" to more advanced ideas like "homogeneity", "symmetry", "letting the parameters to their extreme values", and such) more than on the errors themselves. I'm not sure if it has anything to do with "avoiding negative thinking". Rather it is replacing "don't hit the trees" with "identify obstacles and avoid them as you go".

Ability to detect that something went wrong, a.k.a. "sanity check" is quite important IMHO, so devoting to it some time and developing the corresponding series of exercises seems to make perfect sense. Then, of course, one needs also to develop the ability to actually pinpoint the error, but it is a completely different story.

However the main advice is this: be open to all ideas and feel free to experiment with them, but if something works for you with reliable results, stick to it, and if something doesn't, abandon it no matter what anyone may say.

My advice is to do very little of it, if any. The neophyte learner needs to learn how to do the basic procedures first and then to hone his automaticity. To the extent that you cover errors, it makes sense AFTER the mistake, as a correction. If you try to do it pre-emptively, you are likely to be giving too elaborate of an instruction (remember the new learner is mastering the procedure first, so long winded instructions tax their working memory). You want to give bite sized instruction and intersperse it with successful practice in a gradually scaling up behavior. If you have experienced students, it may make sense to discuss common errors, but that is different from preemptive instruction.

I'd say after a test, makes most sense. There's the benefit of huge investment that the learners have made (thus interest). Also, they likely know the material the best they will (as a point in time). Also, they have strong familiarity with the examples (having worked the problems and perhaps even having a key to refer to, if you provide those).

I have found that with some common errors, it is helpful to have acronyms or pet names. E.g. SF-1 (in red pen). Sig figs or significant figures, -1 point. Similarly maybe with +C on indefinite integrals. (Is there a cute name for that? I do know Jamie Escalante had cute names for common mistakes.) But even here, you probably need to just use correction of the mistakes as the main training method. If you strain too hard preemptively, it won't stop the errors and there are much bigger (new) aspects of the chemistry or calculus problem the trainees are demonstrating, and have not yet mastered.

Probably there are other times when it makes sense to stress error analysis, but they are not probably germane to typical math/school learning. For example, dealing with very risky and expensive operations, like flying an F-14. Even here, you need to be wary as trainees need the situational awareness to grasp the watchouts that are being stressed. Telling me all the things I could do wrong is hard for me to remember if I don't know the basics of flying.

N.B. All this said, if you're doing it and it's working, use your judgment.

P.s. I've been a ski instructor, in a very good ski school, at a destination resort in the Rockies. That example feels very made up education writing to me. First of all, it's rare that you would teach tree skiing (given the legal constraints). If you did, it would only be with very advanced skiers. Also, it's obvious even to 5 year old "never, ever" (neophyte) skiers that you don't hit trees. It's honestly something that you don't have to tell them. Yes, with advanced skiers, you would tell them to pick a line (not "the path"), but this is similar to how you slalom flags or deal with moguls.

• In my mind, forgetting part of a procedure should be treated differently from performing an erroneous procedure. In both cases, I need to reinforce the correct procedure. For example, after discussing constants of integration, I will frequently and intentionally leave off the +C and make a big deal out of it if nobody corrects me. However, if students are using an erroneous procedure, then in addition to reinforcing the correct procedure, I need to avoid reinforcing the erroneous procedure, and so I only give the error as much attention as is necessary to get students to stop doing it. Commented Apr 23, 2023 at 19:15
• +1 for the final paragraph. The tree and skiers thing clearly a made-up analogy and one I would not put too much stock in it. Commented Apr 25, 2023 at 12:08

I think an important step in addressing this question is to unpack the term “error analysis.” There are many pedagogical moves that I would argue can reasonably be called “error analysis.” Here are a few that come to mind, in no particular order.

The teacher presents students with an incorrect worked example…

1. without a label and asks students to judge its correctness.
2. and asks students to find the errors.
3. and asks students to justify why the work is erroneous.
4. and asks students to describe why someone may have made the errors.
5. and asks students to correct the work.
6. alongside a correct example and asks students to draw comparisons.
7. and tells students which errors to avoid.
8. and asks students to describe how they will avoid making the same errors in the future.

A teacher can use more than one of these moves in a single lesson. They can have students work individually or in groups. The error analysis task can be implemented before/after explicit instruction, or before/after an assessment. The teacher can have students inspect their own erroneous work.

It’s reasonable that different implementations of “error analysis” will achieve different effects, and this is without even considering student variability! I think this is a big part of why conclusions in mathematics education literature about “error analysis” or “erroneous examples” are mixed.

To address Sinek’s idea, his examples of thinking of elephants and skiing into trees both involve actions which occur because of a lack of conscious control. But as guest troll pointed out, post-assessment error analysis activities are primarily intended to help students recognize their own erroneous automatic behaviors and develop conscious control.

I do think that error-analysis activities can effectively serve other purposes. For example, prior to explicit instruction, counterintuitive erroneous examples can generate cognitive need, e.g. discussing the Monty Hall problem before instruction on conditional probability.

For these types of error analysis activities, it seems like a good idea to (1) “contain” the error within a specific and memorable problem context and (2) to avoid unnecessary exposure to erroneous procedures which are easily automatized, e.g. Daniel’s example of $$(a+b)^2 \neq a^2 + b^2$$. (I never write an erroneous identity like this in front of students without the not equal symbol or some other prominent qualification.)

Ultimately, you know the effects of your lessons on your students best. Run an experiment and collect some data if you want to test your hypotheses! And exercise your professional judgment.

The idea is flawed.

In the instance described of the skier, the skier going down a slope is in the middle of their performance. To place the math student as the skier in a equivalent position would be putting the math student in a test, or in a competition.

I have yet to see ski lessons that begin with a student already in skis going down a slope or anywhere near the slope.

The ski instructor will definitely be advising a student skier of the hazards prior to the skier ever getting to the top of the hill. In addition to this, a good instructor of any skill/discipline will advise of how to minimize damage upon contact with a hazard.

At best, this lesson should be adapted to the idea that obvious mistakes shouldn't have focus, but the more dangerous ones should. You don't want to hit the trees, or focus on trees, because they're obvious. This is akin to talking to your pre-calculus class about common arithmetic mistakes made in grade school.

To know what hazards are obvious to your students you need to know your students. An easy and lazy method is to assume they've mastered any prerequisite material, so only focus on errors that the new material introduces.

Sometimes there are subtle differences between something being correct or incorrect. In these cases, it can be difficult to teach a student to remain correct with only showing the correct version. Sometimes the differences are best when clearly shown both.

To a skier "following the path" is an obvious task, as your post suggests that there is a clear designated path and all the skier needs to do is to follow it. I highly doubt that to a Pre-Calculus student who is just learning solving trigonometric equations the path is as clear as the ski path. In fact, if the path were clear to the students and all they had to do was to follow it, solving trigonometric equations would not be so difficult. If students make mistakes, it is because the path that they need to follow is not so clear yet, not because they are not focusing on following the path. So how do you clarify the path to the student? At least the skier knows that there is something called "tree" that should be avoided. Now if you want the skier not to focus on the trees and just focus on staying on the path, that's fine. But does a student know what a "tree" is in solving a trigonometric equation? I doubt that. So maybe we should show our students the "trees" first, then ask them to follow the path and not focus on the "trees". In my opinion to achieve an effective teaching, an important requirement is to believe that our students are intelligent and talented. I find it absurd to think that if we show our students possible errors, they will make the same errors. This idea may come across as not believing in students' intelligence.

• If students make mistakes, it is because the path that they need to follow is not so clear yet. I don't necessarily agree. It's important to believe that our students are intelligent, but it's also important to recognize that they are not infallible rational actors. Just recently, I was working with a student and wrote down an erroneous formula because it was part of the notes that they had shared with me, and it wasn't until we moved on to the next problem that I started to feel like something was amiss. Commented Apr 23, 2023 at 20:17