Having recently tutored a number of students through their finals, and given a lot of the same advice repeatedly, this seems like a good time to share a couple recent questions on how to approach learning and doing math. We’ll see one student who wants to stop making mistakes, and another who needs to learn to take chances on unfamiliar problems. They both need to know that mistakes are normal, but survivable!
Tired of making little mistakes
The first question came from India in January:
Hello! I am currently a student studying in high school. I always go fully prepared, I know the concepts, I know the logic, and when I look at questions in the paper I almost always know how to solve them. But its the solving I falter at. I miss little details, either misread a question, or make a calculation mistake – for the hundredth time. Or for example, while doing componendo dividendo, I do it for the LHS but forget to do it for the RHS. For the total surface area of a cylinder, instead of 2πrh + 2πr^2 I’ll do 2πrh + πr^2.
I am tired of it and I can’t seem to improve and I feel like a failure and I feel hopeless. Could you please help me anyhow? I would be so grateful if you took some time out to help me.
Thank you!
If you’re unfamiliar with “componendo dividendo”, which we don’t teach here in America, it’s a property of proportion that we discussed in Proving Proportions, Problematic Products, and again in Proportions vs. Algebra in Proofs.
I answered:
Hi, Deboleena.
I’ll first refer you to what we’ve already written about this topic (which you may have already seen):
How Can I Stop Making Careless Mistakes?
There we offer several lists of things you can try, which probably includes most of the advice I give to students when they have this problem.
(What’s there is mostly excerpts from longer answers, which you can read by taking the links on that page. They require a free account for access.)
But you’re already following one of the recommendations there: “Keep a record of the kinds of mistakes you make.” That’s good!
Many students I work with, for example, have trouble with signs, either thinking wrongly about them, or just copying incorrectly; in fact, in our tutoring center, we had a joke: “When they clean up here at night, their vacuum cleaner bags get filled up with all the minus signs that are dropped during the day.”) Knowing that, I always check signs in students’ work – and in my own!
Based on your specific examples, I think my main advice would be what I say to many of the students I tutor: Write more than you think you need to, and then, every time you write a line, compare it to the previous line to make sure you didn’t miss anything. When you see something in writing, you are more able to see errors than when it is all floating around in your head. In particular, you should write what you plan to do, then do it, and then check whether you did what you planned.
A typical example of this is in solving a system of equations by addition (elimination): I will write next to each equation what I want to multiply it by; then do that multiplication; and then go through each term to see that I did the multiplication correctly. A common mistake is to forget to multiply the right-hand side of an equation; this will catch that.
Also, as I tell students all the time, mistakes don’t mean you’re a failure; they mean you’re human. I make them all the time; I just know how to catch them (usually) before they cause any trouble. My latest slogan is “Half the work of math is fixing your mistakes. And the other half is making them!”
If you’d like to show us examples of specific mistakes you’ve made, there might be more we could say; or if you try our ideas and have trouble carrying them out, that might be worth discussing further.
She had given one example already, so I looked at that:
I’ll make one specific comment, about your issue with using the wrong formula for surface area. I generally avoid merely memorizing formulas, because it’s so easy to forget a number (like that 2), or to pick the wrong formula. Instead, I like to have an image in my mind, when possible, of why that formula is what it is, which makes the details more memorable. We discussed this here:
How Can I Remember Area Formulas?
In particular, the surface area of a cylinder is the sum of the lateral surface (a rectangle with width equal to the circumference of the top, and height equal to the height of the cylinder), plus two circles, the top and the bottom. If I write the formula, make sure what I wrote makes sense, then use it, and then make sure I used the formula I wrote. I’m giving myself lots of chances to get it right!
We might just say, memorize formulas in a memorable way, then use them in a secure way.
Details
Deboleena wrote back:
Thank you so much for your advice!
I don’t memorize formulas; I do apply it then and there but I miss out on writing it (like I mentioned, the 2) and I have absolutely have no idea why. I know what to do, but I somehow skip out on it. This proves to be a menace during exams! I have no problem with speed, I finish my questions before time, but speed without accuracy is nothing.
Just yesterday while doing 35/2 I thought “17.5” but while producing that on paper I mixed it up and wrote 27.5, I don’t think I’m fully focused on what I’m writing. Maybe I *think* I am, but in reality I’m always thinking ahead while doing the sum, “what am I doing next?” I’m tracking that.
Anyways. I’m trying to meditate to improve my focus and concentration and be calmer. I have a major exam coming up in 45 days. I need to get a 100 in it!
Any immediate tips?
I think I’ll get tested for ADHD. Maybe I do have it and that’s hampering my ability to be fully focused?
I replied:
My first thoughts are, first, that since you have no trouble getting work done in time, you are in a good position to follow the first piece of advice from the first page I referred to: Slow down (and take time to check)! And, second, that anxiety may be making it hard for you to do that. Thinking ahead can lead to one thought “tripping over” another. Perhaps you need to give yourself full permission to do one thing at a time. Remind yourself that accuracy is more important than speed.
Students often admit to me that they tend to rush; and it can be hard not to, when they are not sure of the material and tend to be slow. But I remind them that even then, rushing only makes things worse. “When you run, you trip!”
Writing down the wrong thing could suggest a small amount of dyscalculia (the math version of dyslexia), but that doesn’t sound likely in someone who does so well. The ideas you have sound good; and you could ask about other issues (like dyscalculia or anxiety) when you ask about ADHD. I have no experience with such issues, beyond being aware of them, but someone with experience may have some great ideas for you, even if you are not diagnosed with anything specific.
Reported rates of dyscalculia are higher then I would have expected from the description of its (rather severe) formal symptoms; I suspect that many such people just are not in the population I see in tutoring, but also, it may often take a milder form. But it does seem likely that those who work with students with specific disabilities may have good hints for those who have lesser difficulties, so they are worth talking with even if you don’t have one.
I’ll add one more thing. When students tell me about things like this (or when they just make a mistake in front of me), I will often point out that I, too, have some recurring errors, the main one being that when I divide one number by another, I too often write down the divisor instead of the quotient, because that’s the number foremost in my mind at the moment. That’s pretty close to what you did in writing 27.5, having just divided by 2! How do I deal with that? By a habit of reading what I just wrote and comparing it to what I meant. I just know myself enough not to trust everything I do. (Another of my slogans is “Trust yourself enough to try; distrust yourself enough to check.” That one is mostly for students who are afraid to even try a hard problem, but it can apply to all of us.)
Checking is so central to how I work, that my standard procedure sometimes is to deliberately write something I expect to be wrong, planning to check and correct it.
For example, say I want to integrate \(\int\sin(3x)dx\). I know that the answer will involve the cosine, so I just write down \(\cos(3x)\), leaving a space before it. I then check by taking the derivative: \(\frac{d}{dx}\cos(3x)=-3\sin(3x)\). That isn’t right, but if I multiply by \(-\frac{1}{3}\), it will become right; so I fill in the rest of the answer: \(\int\sin(3x)dx=-\frac{1}{3}\cos(3x)\). I find that easier than trying to remember a general formula, \(\int\sin(ax)dx=-\frac{1}{a}\cos(ax)\), and worrying whether I got it right.
We’ve been talking mostly about the “checking” half of my slogan; next, we’ll turn to “trying”.
Unnerved by uncertainty
The other discussion I want to share came from the U.K. on April 1, presumably from a tutor:
My student is working at the very highest level of A level maths. Throughout her career I don’t believe that she has practised or been encouraged to puzzle over questions. She has just been in a position where she knows exactly what to do. But now practising for an A star with challenging past exam questions she cannot accept that puzzling is required, even trying something and not knowing whether it will take you nearer the answer or inform you what different path to take. Any suggestions how I can help? I don’t seem to be able to convince her. Everything I find online just makes it sound so easy …
How can she get this student to try something, when she doesn’t know what to do?
I answered:
Hi, Susan.
I don’t know that I have any great ideas, but I’ll just try thinking through this.
… she cannot accept that puzzling is required, even trying something and not knowing whether it will take you nearer the answer or inform you what different path to take.
Certainly what you express as the goal is something we strongly agree with, namely that you need to be willing to move forward without being sure what approach to a problem will work; we’ve called that “exploration“, and it is definitely an essential part of good problem solving. Life is full of situations where you don’t know what to do for certain, and have to just try things to see what will work. I discuss this with students all the time, particularly in working on trig identities or similar proofs, and in calculus where methods of integration involve trying several “tools” until one works.
In particular, I think of the classic book How to Solve It, by Polya, which talks about various “heuristics” – which basically means “ways to try something without certainty“.
This is the usual context of my slogan about trusting yourself enough to try.
The two courses I mentioned (Precalculus and Calculus 2) give the student a large “toolbox”, with some general ideas about clues to look for (e.g. “If there’s a square of a trig function, consider using a Pythagorean identity”), but no absolute rules. But even in less complicated topics, students may doubt themselves, thinking, “I’ve never seen a problem like this before,” and need to be reminded that they can still take a small step (even without knowing it’s in the right direction), hoping that the next step might be visible from there.
I can imagine some different ways in which a student might develop the problem you describe. Has she never been exposed to non-routine problems where exploration is needed? Has she never (even in the real world outside of class) been in a situation where she had to make unfamiliar decisions for herself? This is partly a matter of growing up! Also, it’s common for students who have always done well to develop a “fixed mindset” (that they are what they are, and can’t risk being wrong) rather than a “growth mindset” (that anyone can grow in ability, by taking chances and learning from their mistakes).
Heuristics are a part of life – and, as I said, of growing up. I don’t know to what extent it’s true that teachers today avoid challenging students too much, but if it is, that is certainly a disservice to the students, who will only grow through challenges.
I’ve mentioned several keywords in my musings here that suggest directions to consider (and pages on our site to search for using those words).
(By the way, do you notice that right now, we’re using heuristics here to solve a non-routine problem with incomplete knowledge?)
In other words, in trying to help a student, we may be modeling exactly what she needs to learn: trying to figure out what will help, when we don’t know exactly what lies behind her difficulty. Sometimes all you can do is ask the student what she’s thinking; and sometimes you just have to “ask” a problem what it needs! Sometimes you just try different ideas to help the student, and sometimes they will see the value of trying things without knowing ahead of time which will work.
I searched our site for some of the keywords I’d used:
Here are some posts that mention various of these ideas, not that any of them answer your question, but in looking at them I’ve been reminded at least of the importance of what you want to do:
Non-routine Algebra Problems (Polya)
John Conway on Thinking and Teaching (last half, on teaching and intuition)
The 24 Game and Kin (puzzles can teach heuristic thinking!)
Proving an Identity in Different Ways (trig identities and heuristic thinking)
How to Write a Proof: The Big Picture (proofs involve exploration)
How Can I Stop Making Careless Mistakes? (mentions growth mindset)
Some of these might help directly, if the student is studying that particular topic; others just illustrate how we try to encourage trying.
So, what ideas do I have for actually helping a student understand that it’s okay to not know exactly what to do, and that uncertainty is an essential step in reaching certainty?
To some extent, this has to be discovered individually; perhaps it will take experience in other areas of life that don’t provide certainty. That might include ordinary (unguided, face-to-face) social situations where the unpredictability of other humans can force you to experiment; games and puzzles that require heuristics to solve; or whatever.
It occurs to me now that one of the costs of the pandemic was a loss of face-to-face connections that develop some of these skills and mindsets. The same is true, as we hear from many sources, of social media.
There are various things I say to students I tutor when they lack confidence to try something they aren’t sure of. Some need harder prodding than others. (I’m thinking of one boy who pushes back a lot when I suggest that something might work, and a friendly reminder of the last time he thought something wouldn’t work, but it did, gets him to try, and succeed.) If the problem is that she thinks “smart people always know what to do”, then maybe some stories of people who didn’t would help. (Here in America, we sometimes tell about Thomas Edison, who persevered through many failed attempts, and said, “I have not failed 10,000 times—I’ve successfully found 10,000 ways that will not work.”) Another thing I do is just to model a spirit of adventure; after solving a hard problem together, I say “That was fun!”, and then tell them that my idea of “fun” is “challenging”. Not everyone likes challenges, but at least tolerance of challenge is essential in life, since challenges will come. And that’s what’s happening when you don’t immediately know how to solve a problem!
Or maybe I’ve misinterpreted the situation, and you need something else entirely! You can tell me more about what sort of situations need to be dealt with, or what sort of intellectual or emotional responses you get.
The other day I added something to my definition of fun, in telling a student to go have fun on their final: Fun is a challenge that you manage successfully! And that’s what I was wishing for her.
Susan replied:
Hi,
Thank you so much for your brilliant reply. It’s just encouraged me to keep going, ‘talking’ to you about it. I’m frustrated with the non-explorative messages that come from teachers and online videos and this student seems to be particularly suffering emotionally.
I think this describes the issue brilliantly:
it’s common for students who have always done well to develop a “fixed mindset” (that they are what they are, and can’t risk being wrong) rather than a “growth mindset” (that anyone can grow in ability, by taking chances and learning from their mistakes)
Thank you
Apparently my ideas were worth more than I thought they might be!