In the Ask Dr. Math service, new volunteers went through a period of training (“internship”), first demonstrating their ability to write effectively about math with some example problems, and then answering actual questions under supervision, discussing their answers with a mentor online before they were actually sent out. Once our reliability was confirmed, we were given “tenure”, so that our answers went out immediately.
To see something of what our goals are, I thought I’d share part of a Guide to Writing Responses, which I believe was first written around the time in 1998 when many of us were invited to join.
After a brief discussion of technical details (how to type math with the primitive facilities of the day, and how to do the writing), there is a list of 12 principles for writing an answer. I’ll quote them, and add a few comments.
1. Make sure your math is correct!
Answer only questions for which you are absolutely sure that you understand the mathematical concepts that motivate the question. In addition, even when you are very sure that you understand the math behind a question, it is important to be extremely careful when responding to questions. Many math doctors have made errors on simple problems. If you are careful to proofread, you can avoid this!
Correctness is goal #1, and always has been.
One of the benefits of being a Math Doctor is having other Math Doctors around you. Each of us can specialize in what we know best, knowing that someone else (probably) can pick up the questions for which we don’t have solid knowledge.
But, yes, we have made mistakes, and though published answers were edited, some of the errors have remained for years until someone wrote to us about them. We have always been grateful for such corrections. We can’t make corrections in the legacy archive, but you can still write here when you find an error, and we can post an errata page.
2. Focus on helping students learn to think in the creative ways that foster a deep understanding of the mathematical concepts lying behind problems.
After having worked with us, we would like students to be able to solve not only the problems that they sent us, but also problems that require similar kinds of thinking.
Make sure that your answer gets at the math that underlies the problem. Avoid giving algorithmic solutions to problems; rather, help the student understand the mathematics that motivates the question. Furthermore, write your answer in such a way that the mathematical ideas and thinking involved in solving the problem are highlighted; after all, it is these ideas and ways of thinking that make math the subject that it is. When you do these things, the student will be able to take his or her newly gained knowledge and apply it to other problems that require the same kinds of thinking; he or she will have truly learned from your response.
Goal #2 is deep understanding, which is far more important than giving answers.
Learning has not really occurred until the student understands the concepts, and can creatively apply them to different problems. So, whenever possible, we want to go deeper than the specific problem that was asked about.
3. Write as clearly as you can.
Remember that many of the students who write to us are struggling with math. Because of this, you should write as clearly as you can when you are writing about math. If the students have to struggle with something, we want them to struggle with mathematical concepts, not with what we mean when we say something.
Goal #3 is communication. Math is hard enough; we don’t want to complicate things.
Many technically-minded people are not comfortable with writing, which is one reason for the training process, to weed out people who aren’t consistently able to do this well. All of us, I’m sure, have written less clearly than we would have liked to, but this is a goal.
4. Explain not only how to do problems, but also why you do them the way you do.
We want students to learn about math and become better mathematicians as a result of having asked us a question. As we all know, you cannot be a good mathematician unless you know exactly why you do what you do to solve problems; knowing only the steps one must take to solve a problem without understanding why one must undertake them will only get you so far.
This is another aspect of #2; understanding involves not only knowing what to do, but why. Math is not done by memorization, but by logical reasoning, which includes identifying when a procedure is applicable, and when something else needs to be done.
5. Let students have a role in coming up with solutions to their problems.
When students ask you how to solve specific problems, help them by setting up a solid foundation from which they can work to solve the problems on their own. It can be difficult to find the right balance between explaining concepts necessary to do the problem and letting students figure out at least part of their problems on their own, but as you work on it, you will get better at figuring out how much to leave to the student.
Interaction is an important part of teaching math; as we say, math is not a spectator sport. We have to get the student to do the work, not just watch us do it.
In those early years there was no efficient way to respond to us (messages came in by email, and were not put into threads), so there were not many extended conversations; but when possible we would try to stop short of giving an actual answer, so that the student would be able to apply his or her own mind to finishing the problem. Sometimes we would ask the student to get back to us, but they never did.
Sometimes we just explain the concepts or give a similar example, leaving the original problem for the student to do; or we might just do the first part or give a hint. How to do this varies from one problem type to another, and sometimes you can’t get around demonstrating all the work.
6. Read questions carefully so that you can tailor your responses to the students’ needs.
Students sometimes tell us what they have tried to do to solve problems, providing information about how far they have been able to get. Use this information to write your answer. We want to address their attempts at solutions either by explaining why what they did was correct, or by showing them where they went wrong.
Ideally, we want to see the student’s thinking, in order to have a better idea what knowledge to assume. Knowing their context (grade level, course, etc.) can also be important. Once it became easier to dialogue, we would more often just ask for such information before giving more than just an initial hint; that is how we often operate now. Sometimes when they respond by showing their work, we discover that their difficulty is in an entirely different area than we expected; and we are able to commend them for what they got right while helping with the error or getting them past a point where they were stuck.
When there is work to evaluate, it is important that we both help them to see their errors, and also give positive help toward a correct answer. Correcting a misunderstanding is more important than merely providing a correct explanation.
7. Look out for questions that have multiple interpretations; when you find them, be sure to make explicit the fact that the problem can be interpreted in different ways.
It is easy to read a question quickly and think that you know what a student is asking; however, always read through questions carefully to make sure that there is only one reasonable interpretation of the problem. If you see that a problem may be interpreted in more than one way, address this in your response to the student. Depending on the question, you will want to respond in a number of ways.
If it seems easy to address all of the possible interpretations, you may do so after you have explained each of them thoroughly. However, it may take too much time to go through each interpretation and one may seem more likely than the rest. You might want first to acknowledge the fact that there are multiple interpretations and then to address one of them. Of course, in this case you should be sure to encourage the student to write back if he or she wanted us to respond to a different question.
Sometimes the number of interpretations is so large and the meaning of a problem is so unclear that you will want to wait to answer the question until you know which interpretation was intended by the student. In this case, it is fine to e-mail the student asking for clarification. To help the student, however, you should include some of the possible interpretations you have come up with on your own. This will help students see where wordings are ambiguous and should thus help the student to reword problems to avoid ambiguity.
I often give similar advice to students when a problem is ambiguous: State your interpretation of the problem before giving your answer, so that it will be clear that you are giving a valid answer to the problem as you perceive it, even if it is wrong according the problem’s intent. By showing possible interpretations of a student’s question, choosing one, and giving appropriate help, we are maximizing the chances of being useful.
8. When possible and appropriate, use students’ questions as a jumping-off point for discussions of related math topics.
If a question clearly leads into a mathematical concept/idea that you find to be interesting, tell the student about it! Even better, ask the student some extension questions that may lead to discovering or thinking about these ideas.
We want to encourage curiosity, because investigating one’s own ideas is great for motivation and for deeper learning. Demonstrating our own curiosity about their topic, or interest in their ideas, can be a way to do that.
9. Write using a tone that is friendly and conversational.
Responses written in a nice and easy tone are more enjoyable to read, and they make math less scary for those students for whom math is difficult.
One of the purposes of the service from the start was to help students become comfortable writing about math. Anything that enhances comfort is good. So one of our goals is to be different from textbooks, both in style and in substance.
10. Be careful with technical language; if you think that there is any chance that the student to whom you are writing will not know a word that you plan to use in your response, either do not use the word or clearly define that word.
Many of the kids who write us struggle with the language used by mathematicians, so make sure that everything you write will be understandable to them. You will probably have to change your vocabulary somewhat so that you use less math jargon. Of course, you shouldn’t avoid technical language altogether; after all, students need to learn math vocabulary words. However, when you do use technical language, provide understandable definitions of words used so that students who don’t know the required vocabulary can follow your answer.
This is another important balance, between teaching important words, and communicating clearly to those who don’t yet know them. It’s also another reason we like to have some background information on a student, to give us a better idea what words and ideas to assume they know.
11. Be supportive and encouraging.
Whenever students get something right, compliment them! Many people’s math problems have more to do with confidence than anything else and a little boost in confidence can go a long way, so be positive. Look at some very encouraging messages.
This is something I try to be particularly careful of in my face-to-face tutoring as much as online. (Actually, all of these principles have defined my interactions there.)
We commonly start off by pointing out the good work a student has done in the work he shows, or even in the phrasing of a question.
Finally, the last and perhaps most important principle:
12. Have fun, and be creative!
If you are having fun writing a response, chances are the student reading your response will also enjoy it. Try out new things with your responses – make some jokes (if that is in your nature) and just have a good time!
Enjoying math can be infectious. (Well, some people seem to be immune, but we can try to expose them.)
We still try to do the same things in our interactions today, and I think these points are useful for anyone doing tutoring, online or off.
For most of these points, several examples were provided in the original document.
Looking for newer examples, it occurred to me to search instead for students’ thanks, to see when they have recognized these qualities in our help. Here are some, listed by principle:
#2: Focus on understanding
The Riemann Zeta Function: Extended Confusion about an Analytic Continuation Now I understand. The beauty of mathematics could be appreciated only when it is understood properly.
#3 Clear communication
Why Does Height Formula Use -16 Instead of -32? Fuzzily getting clearer (a lot) - thanks for getting down to my level.
#4: Explain why
Proof of Derivative for Function f(x) = ax^n Thank you very much! This is very cool, and now I understand why the 'quick method' actually works. Thanks for taking the time to answer my question.
#5: Let them do the work
One Variable in Two Radicals I was going to follow up with one final question for you -- about why the quadratic formula introduces some roots that are extraneous -- and you answered my question before I even asked it! WOW! Thank you for explaining that, and also for not just giving me the answer but for offering a suggestion and letting me work through it myself. You are also a good cheerleader for students. Awesome instructor, Dr. Peterson!
Adding a 6-Inch Layer of Gravel Thank you very much for taking the time to break my problem down into an easier way of managing it. It helped me out a great deal. I understood the problem and was able to figure it out with starting with the easier problem first. Thank you for your time. Have a great day!
#6: Tailor to students’ needs
Isolating a Variable in a Tricky Place, or Raised to a Power Thank you so much, that is exactly what I needed to know. You explained it perfectly and I actually understand it now.
#8: Ask extension questions
Ogive, More or Less A good teacher teaches, whereas a great teacher inspires; and Dr. Math, you have inspired me to learn more. Specifically, I'm going to learn about discrete and continuous probability distributions very soon. :> But I've learnt a lot already. Thanks!
#9: Be friendly
Hockey League Tournament Schedule I really just want to say I can't thank you enough for your time and effort and the very speed with which you responded again. It is so, so gratifying I'm genuinely moved. You don't know me from Adam, as we say - which means you have no reason to be so helpful and so generous, and there's nothing in it for you. These are controversial times when the Internet is creating so much adverse feeling. The privacies and sensitivities of people are so often abused by mischievous and sometimes evil people, hiding behind the anonymity of the computer. To be treated in so friendly a manner and to be given so much assistance on what many people would dismiss as a trivial matter, is positively uplifting.
#10: Define words
How Can the Product of Two Radicals Be Finite? Thank you for taking the time to answer my question so thoroughly. I think I need to eliminate "infinite" from my vocabulary so I don't get confused again! You provided several explanations and scenarios and you MADE it make sense. I really appreciate it and it definitely helped! You're great :)
Accuracy in Measurement Thanks for your reply. It really helped me get over this paradox, and made me feel much more assured towards math. Thanks!!!
#12: Have fun
Can Rewriting P -> Q as ~Q -> ~P Lead to a False Conclusion? You're so nice. We've really learned some new things, and one of us started to get interested in mathematics again.
Is There a "Discriminant" for a Quartic Equation ... in Closed Form? I love Dr. Math. I love what it does. I love how valuable it is -- in my opinion, more than gold. Because while gold can wrap around someone's finger or neck and gleam, Dr. Math helps to improve the mind and mathematical abilities, which I see as priceless, especially when they are as polished as the ones you have. You have shared your knowledge with me; and for that ... and for your input ... and for the extensiveness of your responses ... I am extremely grateful!