In its more than 20 years, we answered hundreds of thousands of questions on Ask Dr. Math, only a small fraction of which were archived for public access. Those that did not make it into the archive were not bad answers, just not deemed to be of broad enough interest, or unique enough, to be worth the effort of editing. (Thanks to “Doctor Tchen” and others before him for taking on that job.) This is one of the features that distinguish us from other math help sites: we focus on giving reliable help to individuals, and put only the best online, where others commonly allow anyone to answer, and make everything they do visible.
The purpose of this blog is to make our work more visible, by calling your attention to particularly interesting examples. The idea for it arose several years ago, as I often found gems while researching for new questions, and wished for a way to call attention to them.
In these first couple posts, I want to share a few random examples of the wide variety of answers that can be found in the archive; someday I will show a couple that are not archived, to illustrate the everyday sorts of work we do. (I only quote excerpts from these pages. You are encouraged to read the whole thing in order to see more clearly what I am talking about.) Some of these may stimulate questions of your own, which you can ask on our Ask A Question page.
Many questions, not surprisingly, come from students who want help with homework. We try not to solve the problem itself, but give hints or solve a similar problem when possible. Other times we work with the student step by step so that she is doing the bulk of the work. Often we have to help students learn how to ask a good question; one of the goals of the site is to give them experience in communicating effectively about math.
In the following example (from 2013), the student did everything right: she stated the exact wording of the problem, showed her work, and stated her issue, which was that the answer she got was not what the book showed:
Composition of Functions: Inequalities in the Domain, Inaccuracy in the Book I am confused. How do you find the domain of composite functions? The homework problem is: Given f(x) = SQRT(x + 1) and g(x) = 1/x. Write a formula for f(g(x)), and find its domain and range. I found the domain of f(g(x)) to be x > 0 or x <= -1. But the answer according to the back of the book is that the domain of f(g(x)) is x > 0.
Then she went on to show her work in detail … except for a crucial step, where she had made an unsupported leap. It turned out that the answer she got, though it didn’t follow from her work, happened to be correct.
In such a case, it was appropriate to go all the way to the final answer, to show what was wrong in Melissa’s work. I did so, but neglected to explicitly point out that the book’s answer was in fact wrong! She wrote back asking for confirmation of that, and I replied by demonstrating how to be sure the book’s answer is wrong (which is not as uncommon as you might think).
... So it looks like the book is wrong (or you read the answer to the wrong problem). Moral: If your own work convinces you that the book's answer is wrong, do what they may have forgotten to do, and check their answer. Sometimes they are wrong!
By discussing what was wrong in the student’s work, and also showing three ways to do it correctly, I provided a thorough example in place of the book’s wrong answer, making the exercise more useful. In other cases, I might just have pointed out that she had to solve a rational inequality and asked her what methods she had learned for them. (I might also have referred to examples of this in the archive.) Such an answer would not have gone into the archive, since it would not add anything useful.
We often get questions from parents about helping their own children with homework; sometimes they just forget details from their own education, other times the child is learning something entirely unlike what they had learned, and they want help to understand it at all. Here is an example from 2001, where Dr. Mitteldorf gave just enough of a hint, along with good advice for parents in general:
Parent Pulling Her Hair Out I'm very sorry to bother you, and wouldn't, BUT am at my wits end. I am helping my son with his homework and we have become stuck on a problem. We would very much like to solve this. ...
The answer includes a description of what we try to do as much as possible:
There are so many things I want to say to you, starting with 'Don't apologize!' We're here to answer questions. We're doing it as volunteers because we enjoy it. I'm glad you wrote to Dr Math and gave me the opportunity to think about this with you. You needn't wait until you're at your wits' end. I'll tell you a story. I used to work in a think tank with other physical scientists. People would come into my office when they were stuck, and I'd "help" them. All I did was to listen to them and nod as long as they sounded confident, and when they started to sound uncertain, I'd say "explain that to me again - I'm not sure I understand it." Well, many a scientist solved his own problem at that point, and I was credited with being a "brilliant" resource. ... Solve it yourself, to be sure. But don't then give the answer to your son - just encourage him to keep trying different things until he gets it. The name of the game is to have interesting experiences exploring mathematics, not to get to the one and only right answer as fast as possible. So we redefine success: success is an enjoyable challenge, with exploration, trying different ideas, and honing of mathematical skills. One more thing: Your hair is quite fine just the way it is. You don't want to teach your children anxiety. Teach them instead to enjoy their explorations.
The parent found the hint perfect, giving the story a happy ending:
It worked EXCELLENTLY! It took 30 nano seconds to do a problem that I had literally been working at for 2+ hours. (Yes, I know it was a bit obsessive, BUT I HATE not solving puzzles. I spent much time going through the books trying to "see" what I was missing, when it turns out I just needed to "see" the problem in a different way.) I had told my son we would go over it later, so no, he didn't "see" my frustration. You totally made my evening! I can put it to rest now and discuss it with my son in the morning (plus the neighbor girl who is also having difficulty with factoring). Have a very wonderful weekend!
I particularly enjoy interacting with teachers, knowing that any help I give them will be multiplied in their students. Sometimes we correct misunderstandings they have developed, or fill in gaps in their knowledge; other times we help them with ideas for teaching (though our main expertise is with the math itself, rather than with how to teach fourth graders – which gives us a different perspective that can be helpful).
Here is an example of this sort of contact, from 2002:
Subtracting and Borrowing in a Column I teach high school chemistry and teach significant figures. This was the problem. 6.9 - 7.92. I worked the problem as usual, getting -1.02, reported to -1.0 with significant figures. A student of mine brought up a point that I had never thought about before, and boggled me. He borrowed as follows 6.9 - 7.92 ---- to get - 2.98 Why won't borrowing work with this number??
Students can lead us in directions we’d never have thought of on our own! That’s one of the fun parts of answering these questions. Here, the context was significant figures, and the teacher’s sense of the issue was that borrowing was the problem. But I realized that the real issue was something else entirely: the fact that the answer was negative. There was nothing difficult about this; I was just a fresh pair of eyes on the problem. But I was also able to relate what was happening to place value issues, and even to computer representation of numbers.
Next time I will continue randomly sampling different kinds of questions, moving away from the academic context to look at real life questions, and some of the questions people can raise just out of curiosity.