# How do you coach students who often make small errors?

Some students are prone to making small calculation errors. Not errors in understanding, but errors like adding or multiplying integers incorrectly, or dropping a negative sign. Unsystematic errors in the terminology referenced here. Sometimes there's an obvious cause to this, like if the student writes sloppily and literally loses negative signs in their calculations or can't read their own handwriting, or if they're rushing through the calculation.

But suppose I've got a student who writes neatly and performs calculations carefully and methodically, but still makes a ton of small errors. How do I coach this student? "You need more practice" is probably true, but can't I be more helpful than that?

• I see something like this where a student taking an integral will perform every step of the advanced mathematics correctly but then reduce a fraction incorrectly or flip a sign. I don't think it is because they are at the end problem in my case, but because they are doing arithmetic. I suspect one cause is there are multiple and dissimilar thought processes and their focus has switched to a higher order conceptual mode, leaving number sense behind. I encourage students to be metacognitive about this and intentionally switch back and forth between the bigger picture and the details. Apr 2 '21 at 11:45
• You might find resources by looking for ways of supporting students with dyscalculia. Apr 2 '21 at 13:14
• How often is “often”? As a % of calculations is it more like 1%/5%/10%/25%/50%? Apr 2 '21 at 13:36
• @MikePierce Yes. If it is on the lower end (10% or less = 1 or 2 careless errors per test), then I think encouraging the student to check their work (by back-substituting) could immediately help them determine /when/ the situation has occurred and then they can look line by line for the issue. If it is more often than that, then there is no quick fix; either the student does not know arithmetic well enough (needs more practice) or there may be a deeper dyscalculia situation occurring. Apr 2 '21 at 14:05
• @Steve Oh yeah this beyond the 1 or 2 careless errors threshold. So it sounds like I need to look in dyscalculia. Apr 2 '21 at 14:09

### Ask the student to "talk through" their calculations

Having a student verbalize their calculation may force them to pay more attention (or a different kind of attention) to their work that causes them to catch the errors as they make them. This feels very related to rubber duck debugging. At the very least, if you're working with a student one-on-one, if they don't catch their own error then listening to them verbalize their calculation can help you catch their error as they make it, and maybe give you some insight into what thought led to that error.

– J W
Apr 2 '21 at 15:27
• I used to review my own calculations sort of "meta". That is, I would star each line which had the potential of carrying a sign error. I got used to doing it on the fly. This made me catch several errors as they happened, and more when I was proofreading. I would never have survived matrix calculus without it, seeing as I am not a very rigorous math practicioner. Apr 5 '21 at 9:49
• I've always struggled with this. Sign errors and factors of two. Talking through is the only technique that works for me. It assumes that time is not of the essence. Ideally, ask me to break down my calculation further. Ask me to explain it in detail, to the point of "this is the distributive law a(b+c) = ab + ac with a = _, b = _, c = _." Like I'm telling a computer how to do it. My problem was I'd recognize a sub-part of a problem as a known, solved thing and just stop paying as close attention because I knew it was conceptually boring, even if it was necessary to getting the right answer. Apr 5 '21 at 14:09

I used to have this problem. What helps me more than anything is:

• Solve it two different ways if you can and make sure they agree
• If you are finding a general formula, test it on some examples
• If neither of the above are possible, re-read every step of your work with an attitude like it's trying to sell you a used car.
• Adopt the useful exaggeration that any unchecked step is probably wrong.
1. When there is an obvious way to check the final result, don't give partial credit to students who make a mistake and fail to catch it. If they're solving a system of equations, they should be checking by substituting back in. If they're integrating, they should be checking by differentiation. Do give partial credit if they do the check on an exam, but run out of time and write on their paper that they know it failed the check but didn't have time to track down the mistake.

2. Assign problems in which dimensional analysis is relevant, and make this a required part of the problem. Do an example early in the semester where you demonstrate the technique, and refer your students to a written presentation of it. (College STEM majors should have seen it multiple times already, but could probably use the review.)

3. Require sense-making.

4. Encourage the use of computer algebra systems on homework, when appropriate, but discourage its use in inappropriate situations where an expert would not resort to a CAS. The open-source CAS Maxima is free.

Example: A sample of $$N$$ atoms of a radioactive isotope decays over time according to $$N=Ae^{-\lambda t}$$. $$N$$ is large enough that we can treat it as a continuous variable. (a) Infer the units of $$A$$ and $$\lambda$$. (b) Find the rate of decay, in units of decays per second. (c) Check that the units of your result make sense. (d) Check that your answer makes sense in the limit $$t\rightarrow\infty$$. What is the interpretation of this limit? (e) Suppose that we do such an experiment using a larger value of $$A$$. Interpreting your answer from part b, what is the mathematical effect of increasing $$A$$? Does it increase the rate of decay, or decrease it? Now compare this mathematical trend with what is expected physically. Does it make sense?

I have a bunch of problems in this style included in my OER calculus text. The book describes dimensional analysis in sections 1.7.3 and 1.9.

Dimensional analysis is particularly powerful because if you make a habit of doing it reflexively at every line, it tends to catch mistakes immediately rather than at the end of the problem.

• Apr 9 '21 at 19:26

Teach methods for verifying results.

Often once you have an answer, there are simple calculations to check its correctness. For example:

• Numerical calculations: round the results and compute backwards. Example: if you get 1234 * 56 / (789 - 987) = 349, compute e.g. 300 * (700 - 900) / 50 = -1200 which is not close to 1234, because of the dropped minus sign.
• Solving equations: substitute the result back to original equation, both sides should have same value.
• Simplifying equations: test it on a few values.
• Integrating a function: take the derivative of the result.
• Verbal problems: rephrase the problem backwards. Example: "Cities are 100 km apart. Car 1 travels 80 km/h, car 2 travels 30 km/h towards each other. How long until they meet?" If you get answer of 1 hour, rephrase as "Where are the cars after 1 hour?".
• Geometry problems: check basic identities. For example if the diagram has triangles, verify the sum of corners is 180 degrees.

It's worth noting that the same methods will also work for testing individual parts of a long problem, in order to find where exactly the mistake occurred.

### Allow and promote using a calculator

This is one of the recommendations listed for accommodating students with dyscalculia, but this could help whether or not a student has dyscalculia. It's not really coaching and it doesn't address the underlying source of errors, but at the very least it will keep those arithmetic errors from cropping up and getting in the way of conceptual understanding. On the assessment side of things too, allowing for calculators will keep those small errors from giving a skewed measure of a student's ability, especially in the world of online computer-graded assessments.

• This is like suggesting a drug addict to continue using heroin because he is used to it. Apr 3 '21 at 16:18
• Sorry, I must completely disagree with the 'use a calculator' attitude. That simply promotes reliance on the machine with no understanding of whether their final result is correct. Someone who makes the sorts of errors described in the OP seems equally likely to mistype a number or hit plus when they mean minus and without that basic understanding they won't realize the answer they get doesn't make sense. Repetition is, I suspect, the only way to improve (if improvement is possible, I do admit there are some for whom no amount of practice will significantly improve their math ability). Apr 3 '21 at 16:32
• Is the low grade really a skewed assessment of the ability of a student who can't carry out basic arithmetic ? Or is it a timely warning to them not to neglect the basics, while learning more advanced maths ? Would the former interpretation reflect a belief of the instructor's, that the basics don't matter ? Young people are very quick to pick up, and adopt, even the subtlest of such biases. Apr 4 '21 at 0:04
• @Simon ... it depends on what I'm teaching them? Like, if I want to assess their understanding of the arithmetic of exponents, and they say $(x^8)^7 = x^{65}$, this is wrong, but they do know to multiply the numbers. Their error that $8\times 7 = 65$ triggers a false negative in my test. This could've been avoided had they used a calculator. Apr 4 '21 at 19:07
• @Mike Pierce I am old school, and I prefer not to divide the task into many subtasks, to be assessed separately. Of course sometimes my superiors in their great wisdom have insisted that I do so, in order that I may create, and gather data on, such instruments of the Devil as Course Learning Outcomes, Key-Stage Progress Indicators, Attainment Matrices (have I invented that one ?) etc. I believe that the data thus created is very useful to said superiors for their own promotion applications and periodic self-evaluations (which seems strange since it is not work that they did themselves). Apr 5 '21 at 19:35

I believe that the majority of students with this problem are doing what is called "end-gaining" in the Alexander Technique:

https://www.hilaryking.net/glossary/end-gaining

I'm not an expert, but I think that in this case, it means that they are overly focused on getting to the end of the exercise, and they have a mistaken belief that the means for getting there are less important than the end result.

They might also be unconsciously reflecting a disdain for "easy" maths, such as arithmetic, that they have picked up from their educators. Or worse, consciously cultivating an "Absent minded Professor" act, by which they hope to demonstrate their brilliance ! ("I don't have time or patience for that middle-school stuff").

Apprentices in trades can't afford to neglect the basics, and are not allowed to, by their mentors. I believe that we also should (kindly and patiently) enforce proper attention to the "boring details" also. Often this can be achieved just by example.

• This doesn't answer my question of how to coach such a student. Apr 4 '21 at 0:54
• Maybe throw that link in a comment though because its a valid idea, but at least my student that motivated this question doesn't fit the description at all. Like, I asked them to clean up some rational expression (practice rules of exponents) and after checking in on them a bit later, they'd tried to clean it up twice, taking different approaches each time, and started evaluating the original expression and their cleaned versions at specific numbers to see if they matched. Apr 4 '21 at 0:58
• I downvoted this because I don't think it answers the question, although I agree it would be a very valuable comment. Apr 4 '21 at 19:55
• I suppose what I intended with this answer was to convey that I believe that the principles of the Alexander Technique itself provide answers to this problem. However, that is a lifetime's study, and I don't know if it can usefully be condensed into a paragraph here. Sometimes answers to questions are like that. "I think it's a long story, and I think it begins in such and such place". Apr 5 '21 at 19:30
1. As far as "just practice", you shouldn't trivialize this or think of it as so simple as yes, no. Your job is to train the trainee. Many errors are common and repeated. So? This is not so different from sports or music. Develop a set of drills, games, etc. Try different things with different people.

The problem here is not so Boolean as math itself. People are not OR gates. I don't mean to come down too hard on you, but I just want to emphasize that this is a big topic and is central to being a TEACHER as opposed to a creator of math. You need to wallow around in there and try different things (worksheets, books, Kahn Academy, purchased games, competitive drill, etc., and see what seems to work better with different students.

And no, there is no simple square peg for the hole style answer from YA or SE or whatever. It's a big topic. And yes, it may take time and yes some students will struggle more than others. Teaching people to ski the bumps parallel is not instant gratification either. Math is tricky to master, for us organic brains. So are languages, music, sports, and work.

2. Have your selection of problems include some easier ones (less steps). This is especially important at the beginning of a session and with subjects that are more error prone. By having a decent fraction of the problems that are simpler, the trainee can get some experience at solving at least some problems correctly and not become discouraged. Perhaps he also gets some confidence from the simpler successes and incorporates them in longer problems. Of course, there should be some selection of problems that test the trainee with the multistep calculations with which he struggles. But don't make the mistake of doing 100% like that. (Note also that this is in the OPPOSITE direction of teachers who want to emphasize "project style" homework assignments.)

3. Whenever a problem is missed for a dumb mistake, make the student repeat the entire problem (after finishing it) with a fresh sheet of paper. It gives free drill on the concept itself and is a light "stick" to encourage doing it right. It is also more likely to be easy for the student (i.e. reasonable progression) rather than an all new problem or the same problem after a long duration away. This may seem hokey, but it really works. Near term feedback.

4. Do NOT verbally minimize the importance of calculation errors. I.e. "Oh it was just a dumb mistake." Treat it more like tennis where the ball is in/out and it doesn't matter how pretty the stroke. And I'm not saying you in particular do this, just a general precaution. I would be like this all the time. But in this case especially (where the problem is errors), one should not accept the pattern which needs to be trained away. And I'm not saying to crucify the kids either. Just eschew saying "it's OK". Since it's not.

5. I was concerned in comments that you didn't want to bother estimating the error frequency. You are looking at a practical phenomenon. One having to do with human performance. Not Euclid. One of the ways to get your hands wrapped around a soft topic like this is to start by making simple estimates (e.g. frequency) and to segment the errors into different types. And yes, I do think knowing this will help with devising solutions as well. For one thing it gets you involved in studying the phenomenon more.