Dave Peterson, Ph.D.
From time to time, a student will write to us asking for advice on studying, rather than on math itself. As either successful students, or teachers, or (quite often) as adults who recall overcoming difficulties in the past, we have some good advice to offer. Today, I want to look at three answers, by three Math Doctors, over a span of more than a decade, all answering the basic question, How can I stop making these careless mistakes? In each case, the answer is long and I will only summarize the main points; to get the full benefit, you will have to read the whole answers.
The first is by Doctor Ian in 2002:
How to avoid careless mistakes? I have tried do as many problems as possible, but mistakes are constantly made just because of carelessness!
Doctor Ian makes three main points, each illustrated by detailed examples that I will omit:
1. Realize how important it is
Once I became convinced of the importance of what he calls ‘the habit of correctness and precision’, I found that I started adopting it quite naturally, without much effort at all.
2. Don’t do too much in your head
Write more steps than you think you need to, so that you can see what you’ve done. (My personal version of this is: Until you write something, you can’t tell whether it’s wrong; writing clarifies what you are thinking.)
Having said that, it seems to me that many, if not most, of the careless mistakes that we see here at Ask Dr. Math are caused by trying to do too many steps at once. …
A good rule to keep in mind that you can’t make mistakes fast enough to get a correct answer. :^D
3. Always check your answer
(That is, make sure it really does what you were asked to do.)
I also make it a point _always_ to check my answer, after I think I’ve found it. At first this is something you have to remember to do, but after a while it becomes natural.
In addition, doing a check with actual numbers sometimes clarifies what you should have done with variables.
4. Translate “word problems” into equations in small steps
A second kind of careless error, which sort of falls into the same category, is caused by translating story problems too quickly into equations. …
5. But there’s a trade-off:
The less you write, the more easily you can make a mistake, but the more you write, the more places there are to make a mistake! (I just said this to a student yesterday!)
On the one hand, working ‘in place’ is an easy way to get sucked into making careless mistakes; but on the other hand, the more times you copy something, the greater the error that you’re going to copy it incorrectly. With a computer, you can simply copy the old line and change it, which is what I’ve done here. Without a computer, you have to use your judgment about whether working in place or copying is more likely to cause a problem. To make that judgment, you have to have some idea about the kinds of mistakes that you tend to make, and how often you make them.
6. Keep a record of the kinds of mistakes you make
Which leads to my final recommendation, which is that you might want to keep a notebook of the careless mistakes you make. Keeping track of them would allow you to observe patterns, and figuring out what you’re doing is the first step towards changing _any_ kind of behavior.
The second answer was written by Doctor Rick to a different student just a few days later:
No matter how hard I try it’s not good enough. I understand the material – it’s the other stuff, like when it asked me for the range, domain, and inverse of ordered pairs, I just put the inverse because I misread the question. Or like when I had the right answer on scrap paper but I left off part of the answer when I wrote it on the answer sheet. It makes me feel really stupid.
I’ve had this problem with careless mistakes since 6th grade, but it’s getting worse. Proofreading my work helps very little and sometimes I don’t have time when I’m done with the test. What do I do?
Doctor Rick has two main suggestions for this student, who clearly is a diligent student with good observations of his own.
1. Learn from your mistakes
(I ask my own students to rework any problems they get wrong on a test and turn it in for partial extra credit; you should do this yourself even if there is no external reward.)
You have observed some specific kinds of mistakes you make, and that’s a great way to start. One step in problem solving that many people forget about – even after checking your work, which is easy enough to forget – is to look back on what you’ve done and see what you can learn from it. Sometimes you see something positive that you’ll be able to use again – a trick that worked, or a pattern you saw (“when I see this, I can try that”). Other times, as in your case, you see something to avoid next time. The question is, how can you avoid these sorts of mistakes?
2. Make a habit of writing what you are asked to find
(I commonly list the “givens” and the “goals”, with blanks next to the latter; I may also underline these things in the original problem.)
You say that you misread a question, so you didn’t give all the answers that you should have. This is a reminder that another important step in problem solving – the first, and sometimes the most important – is to ask, “What am I supposed to find?” Try making a ritual of starting a problem by listing exactly what you are supposed to find. Then when you finish your work, write each answer next to the list, or at least check off your list as you copy the answers. This will also solve your other observation: that you forgot to copy all the answer from your scratch paper.
Develop a growth mindset
Finally, in 2016, Doctor Floor gave a very helpful answer to a long and thoughtful question:
Our son, a high school junior, is currently taking Advanced Placement BC Calculus. He has always excelled in math (and all other subjects), and never had to study much, because he easily understands concepts. He has, however, always had a tendency of making careless mistakes.
This year this tendency has become a particular problem, with his grades suffering for the first time. Part of his tests are multiple choice — no calculator allowed. Here, he does not have to show any work. But this part needs to be turned in prior to starting on the next section, one where calculators are allowed. On the calculator section, he does need to show his work; and the brevity of the short answers often belies the many intermediary steps they required. …
My son’s teacher says all his mistakes have been careless ones, not conceptual ones: he makes simple calculation errors, or misreads questions, or omits units, or runs out of time, depriving him the opportunity to check his work.
Doctor Floor lists three observations:
1. Smart kids often have surprisingly poor strategies
Smart kids often have learning strategies that seem lazy or careless, due to the fact that they haven’t been really challenged in younger years. Because of this lack of challenge, there has never been any motivation to develop learning strategies or solving strategies. Easy tasks pave the way for good grades, and the cycle reinforces itself.
But at some point in a school career, relying on talent alone turns out to be not enough. Even worse, teachers often think that by high school, high-performing kids must have good strategies. …
So the key is in his homework. Do not only complete homework, but *review it.* Learn from your mistakes. Even when you do it correctly, wonder if you could have done it smarter, or quicker. If you encounter a trick or novel thought, make a note. Be concentrated and targeted in your review.
2. Too much stress holds you back
People cope with this very differently, so consider these as only the most general of observations:
- Be confident. If you know you are well-prepared, there is no need to be stressed. …
- At the same time, be realistic about what to expect. This is particularly important if the test turns out to be more difficult than you thought, or time pressure is higher than you thought.
- Force yourself not to think of any consequences while taking the test. Just take your test, and stick with taking the test. Other thoughts will only break your concentration. Prepare ahead of time so that if you do lose concentration, you already have a way to re-focus and get back on track. …
- If you can show your abilities in a test, that is the best you can do.
3. Mindset matters
Quite a few kids have a mindset that holds them back. It is called fixed mindset, and comes with the thought, “It doesn’t really matter if I do the homework or not; either I will understand it or I won’t.” Often kids who are labeled as “smart” or “intelligent” develop such a mindset. These kids think their intelligence is fixed, learning is understanding, and that developing intelligence does not exist (e.g., by homework).
By contrast, a “growth mindset” posits that developing intelligence is possible. Research shows that there is indeed a correlation between mindset and intellectual development.
The parent responded:
We can’t thank you enough for your response! It was spot-on!
I feel all your points are excellent and will be very useful. I can’t wait to share your response with our son (he came home with the flu yesterday), because I think he will now understand the underlying issues, make the adjustments needed, and as a result cope better. The difficulty will be for him to accept that he needs more practice even if he already “understands” concepts, and to figure out how to change his habits.
While we were familiar with “smart kid” issues and the danger of not being challenged, we had not anticipated that this would become THE instant when everything would back-fire.
As you can see, there are a wide variety of ways in which careless errors can arise; each of us has to be aware of our own personal issues, and make a strategy. I hope these three discussions will help others.
I’ll add something I’ve been telling students recently, which summarizes many of the points made above: In order to solve a problem well, you have to
- Write what you thought.
- Think about what you wrote.
- Then fix it!