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This question might sound vague, but I'm really just looking for particular examples that worked for you.

From my experience, it seems like a large portion of "weak" students remain weak despite their best efforts. I understand that part of this is due to things that have happened earlier on in their lives. Perhaps they didn't learn a fundamental subject well enough and were passed anyway. Perhaps they have poor study skills that haven't been corrected: copy notes mindlessly, look at solutions before attempting the problem, don't memorize, etc.

Besides just reminding them and showing them by example, what can be done?

In particular, my question is this: What have you done in your classroom that has helped instill good study habits or good habits in general. Or alternatively, what have you done to break bad habits?

My go-to weapon is office hours. I can talk to individual students about their study habits, ask them to work problems in front of me, etc. but it seems very inefficient to have to deal with students on a case-by-case basis...and hard enough to get them to show up.

Is this a lost cause or can you teach an old dog new tricks?

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What level of teaching do you hope to find and answer to this question? I feel as though the answer can vary widely depending on the grade level. – Andrew Sanfratello Apr 24 '14 at 21:02
Yes, I agree. I suppose I would like to focus on Secondary level(High School) and Undergraduate students. – Carlos Apr 25 '14 at 2:35
@Carlos just a stackexchange tip -- when you accept an answer to your question this quickly, it has the effect of slowing down further answers on the question. In the future, consider letting the question stay open for a couple of days before choosing the best answer! – Chris Cunningham Apr 25 '14 at 16:30
Certainly! I only accepted an answer because there's not one set answer to this question. I'll keep the questions open for longer in the future though. Thanks. – Carlos Apr 25 '14 at 16:37
up vote 9 down vote accepted

One thing I have done is to have taken the courses I have helped with (Teaching Assistant, Course Assistant, Research Assistant, Instructor) very seriously. A summary of my approach to TAing and office hours (your "go-to weapon") can be found in an earlier MESE post here.

Of the approaches mentioned there, one that I would like to re-emphasize here is the importance of asking that students who would like assistance:

  1. Give their questions due consideration before asking them, and

  2. Indicate how they are thinking about their questions as a part of the asking process.

Some of the so-called bad habits that you cite can be addressed to a reasonable extent by ensuring that students are not viewing you, their teacher or even their peer, as a crutch, but rather as someone to help them in their struggle (ideally a productive struggle, cf. MESE 2) to learn the material.

Here is an actual email excerpt from a graduate student in mathematics education (previously a kindergarten teacher!) who decided to take a course in Topology for which I was the TA. (For the sake of clarity, I added in the TeX below.)

Question: Are basis elements in $\mathbb{R}_d \times \mathbb{R}$ of the form $(a \times b, a \times d)$?

Here is why I'm guessing basis elements in $\mathbb{R}_d \times \mathbb{R}$ are $(a \times b, a \times d)$:

•$\mathbb{R}_d$ is the discrete topology. The discrete topology is the collection of all subsets of $X$, which means all subsets are open, which means $\{p\}$ is open for all $p \in \mathbb{R}$.

•For $\mathbb{R}$ we have the standard topology, which means open intervals $(a,b)$.

•So the basis of $\mathbb{R}_d \times \mathbb{R}$ is $\{p\} \times (a,b)$ or, written another way, $(a \times b, a \times d)$.

Is this correct? Is this completely wrong? I'm doing #9 p. 92 [in Munkres' Topology 2e] and if the above is correct, then the dictionary ordered topology on $\mathbb{R} \times \mathbb{R}$ and product topology on $\mathbb{R}_d \times \mathbb{R}$ have the same basis elements.

I include the above excerpt not for the specifics, but rather because this student initially overwhelmed me with questions. Eventually I asked her to include her thoughts - even if wrong - whenever emailing me. In the above example, note that she includes a guess and her reasoning. In this case, they are pretty correct; moreover, from around this time, the volume of emails dwindled substantially as the student was able to gain a fuller sense of agency.

(The question of whether Topology should be taught in Mathematics Education programs is one I leave to another thread...)

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Thank you. What I'm really trying to get at is "What" exactly is the best way to get students to do these things. Good students might already come to office hours and good students know how to best phrase their questions. However, what do you do to help poor students become good students. In your particular example, how did you get that student to start sending you those types of email questions? (That's great btw. I usually just beg my students to email me with questions and usually those who need to the most don't) – Carlos Apr 25 '14 at 2:45
@Carlos My experience is that students who come to office hours and/or send emails are often at one of the endpoints of the totally-lost to want-every-detail spectrum. I think probably by establishing a critical mass of people who came to my office hours and/or emailed me, I was able to reach more of the people in between these end-points. I'm sure word of mouth helped, and I think the various docs I posted online (notes as in my TA link above, practice exams, etc.) were also encouraging. Attendees were grad students often with teaching experience, though, so perhaps our situations differ... – Benjamin Dickman Apr 25 '14 at 4:04

Give frequent quizzes.

When I say frequent, I mean every day.

The quizzes should be very brief: one basic problem from the lesson of the previous day. The grading will not be too onerous if the quiz has just one basic problem.

Give the quiz at the start of class, after briefly answering questions.

Make the point value of the quizzes very small. Drop several of the lowest quiz scores at the end of the term. No matter how small the point value of the quizzes, students will take the quizzes seriously.

In order to instill good habits in students, you have to first decide exactly what good habits you want your students to have. Surely most of us agree that the best habit is for students to keep up with their work, and the worst habit is to fall behind and give up. Frequent quizzes help to instill the good habit of keeping up with their work.

Assigning homework is not enough by itself. If you assign homework but don't collect it, the good students will do the homework, and the other students will not. If you assign homework and do collect it and grade it, the good students will do the homework, and the other students will cheat off the good students. So in either case, the good students do the homework and the other students don't.

By giving frequent quizzes, more students begin to realize the value of doing their homework and keeping up with their work. And it is not easy for students to cheat on quizzes, if you are a good proctor.

No technique works perfectly to instill good habits in all students, but giving frequent quizzes is very valuable.

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I make it a particular point for first-semester college students to teach them how to learn effectively. I do something similar for second-semester students especially if most of them were not in my class the previous semester. For a precalculus class, I may stress the points below throughout the semester. For a Calculus II class, I adapt myself according to their performance. I should note that when I say I teach them to learn effectively, I recognize that it took me a long time to learn how to learn and it might take most of my students a few years, too.

I have several things I try to tell them at opportune times. When possible, I tell it through a story. It helps to have an interesting roommate in college, or to have made mistakes yourself when you were a college student. I usually say such things at the beginning of class, and limit it to one thing per class. Some of the things I say are appropriate for everyone and some moreso for a student who is struggling with something. In the latter case, I might say those things only in person. It's good to have a balance of positive and negative examples. Positive seems better, but sometimes a warning example is the best you can do, especially if a few students might might recognize themselves in the story.

The important thing is to give advice when the students are receptive to it. Sometimes a student will ask something in class that is a perfect lead-in to one of the points. A few of the topics are meant to prepare the students for advice. For instance the "work versus effort" discussion and the biology of learning below establish terms for discussion I can use throughout the course. One of the times you will have their greatest attention is the day you hand back the first test. (For that reason, I think it is good to have an early test, around three weeks into the term, while there is still time for the student to adjust.) It is important that I have the terms in place before I have that conversation.

Since I am trying to get them to change their habits, it takes a lot more than just telling them what to do and what not to do. It helps to get them to recognize that their habits need improvement. It also takes a lot of reinforcement.

Below, I have put the words I say to the class or student in quotes, more or less as I might say it. Occasionally discussion or comments will appear outside the quotes.

  • Work versus effort. "In physics, work is the product of the component of force applied in the direction of motion. For instance, when you move a heavy object, the work done is proportional to the distance the object is raised. If you carry a 100 lb. object across the quad, it takes a lot of effort, but you do no work if the object is not raised. Learning takes work and effort. You want to raise the level of your understanding standing. You want as much as possible your efforts to be efficient and result in work accomplished." Not all my students know physics, so this needs a brief explanation such as I wrote above. I usually tell this a week or two into the term and well before the first test. Then I remind them right after the first test before they get the scores back. "If you find that your performance wasn't as good as you wanted, you might consider how effective your study habits have been. What you did, did it get the work done? If you have questions or concerns, come see me. I know lots of good strategies for studying and can help you."
  • Learning takes work. "Learning alters the neural networks in the brain, and the alteration takes energy. It really does take work in the sense of physics." This might be what I say the day after the work versus effort story.
  • How learning happens. "When a neural network is formed, it is connected to other networks, other knowledge. There is a process of forgetting that naturally occurs in which the new network becomes less likely to be activated unless it is used. The more it is used the more easily it is activated. More networks form around it as our understanding deepens. One of the more powerful ways to reinforce this new knowledge is when it is used to create motor activity." Some teachers are creative and make up dances or other physical activity. "Writing out the solution to a problem is motor activity. So is illustrating the solution with a graph."
  • Practice. "The primary way to learn mathematics is to practice solving problems." I tell them several times. I can connect it to the description of how learning happens above when appropriate. "Once you learn how to solve a problem, practice solving problems the way you will be tested -- without resources, writing complete solutions. When you don't need to look things up, you will know that you know how to do it."
  • Recall. "The effort to recall reinforces knowledge and makes it easier to recall it. Trying to recall it in different situations helps, too. You can trying recalling things on the way to the cafeteria, say the derivative rules," or how to solve a type of problem, or a proof, etc., according to what's appropriate to the course.
  • Memorization. "A carpenter does not plan to build a house by saying that whenever a hammer is needed, he or she can run down to the community center and rent one. If you're going to build solutions to problems, you have to have the appropriate tools at hand. You'll have to decide what needs to be memorized -- I'll give you hints when I think something is important. In mathematics it is much less than, say, in foreign language or some science courses. The best way to do it is in short, frequent, intensive intervals; then less and less frequently. Pick two times each day some hours apart and practice, with flash cards if you like, for fifteen minutes."
  • Memory and understanding. "Memory is unreliable, at least my recall of facts is. It is easy to switch the facts around, get a minus sign wrong, and so forth. There are a couple things you can do to help. If you understanding why, then that can be used to check. Another is to learn a way to check." (For instance, for the sign of the derivatives of sine and cosine, in the first quadrant, the sine is increasing and the cosine is decreasing.)
  • Repetition. "Sometimes it is more important to repeat a problem than to work a new one." When a student has struggled to work through a problem, I say, "Now what I would like you to do is later tonight or perhaps tomorrow, work the problem again and see if you have learned how to do it."
  • Knowing that you have learned. In response to when a student says, "I don't know why I did poorly": "Did you work the review problems alone and unaided by the textbook and your notes?"
  • Being prepared for the test. (1) "What do you think is the most important thing to do to be prepared for the test?" (Wait for their answers. You'll get an insight into some of your students, and generally you find out they know half the things you would tell them -- at least, they know what to say.) My answer: "To show up. You're not prepared to take the test if you're not there." Then I work back from that to the things it implies, such as getting a good night's sleep, etc. (2) "Be prepared to take the test the day before it is offered. If you wake up and, nightmare!, the test was that day, you should be able to do a good job. If for some reason there is something you still have trouble with, then you have another day to figure it out."
  • Help each other. "Forming habits is easier when your friends are doing it, too. And learning, especially learning difficult material, is easier when your friends are doing it, too."
  • Learning how versus learning to study. It is relatively easy to learn the words of the answer to the question, "How should you study?" It is harder to learn to make it part of one's habitual practice.
  • Positive attitude. It is particularly important that the teacher have faith that the students can do it. Students can sense sincerity. Forming or breaking habits is difficult and the student will have to do it for himself or herself. It took me until junior (3rd) year of college to feel that I had formed good habits and knew how to learn effectively. I tell that to my students. I emphasize that they need to be patient with themselves and they will improve. If they continue improving, they will become excellent.

I started thinking about these things several years ago, when a good student, one who had won a scholarship, came to me and said she was surprised by the questions on the test and they were not what she expected. I said, "Well, let's look at the test. Maybe I made a mistake." So we went through the test and quickly summarized what each problem was about. Then I said, "Now, let's look at the syllabus." It corresponded exactly with the test. She said, "Wait! It's that's simple?!" It made me start paying closer attention to what the students do not know about going to college/university. Last fall, another student, even more highly qualified, earned a D on the first test in Calculus II and came to me for advice. On the next test, he made only one minor error. When I returned the test, he exclaimed out loud, "Oh my gosh! Studying pays off." My colleagues often have a "Well, duh" response, but to me it shows just what the teacher needs to do.

Finally, let me recommend again the physics-based description work versus effort. Such gimmicks often have varied success in the hands of a variety of people, but this one has been effective for me. I do not get far fewer complaints about how hard the student tried; rather, they say, "I put in a lot of effort, but I still did bad. What can I do to improve?" First, I think that is where their attention should be. And second, asking for help is so much more pleasant, both for me and for them.

Helping people change their behavior requires patience. I know the above helps some -- I would say most of my students this year -- and would help all. There are always students struggling with personal issues. A few have said it helped with the rest of their college career.

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The causes are probably a hodgepodge of all of the above, and a few more. First step is clearly to find out the most important factor in each case... and that takes one-on-one time.

Is special tutoring (even in general math) available? If so, make sure the students go there.

Suggest/encourage/enforce studing habits:

  • Take a look at the notes after each class
  • Don't try to cram three days before the exam, it just won't work. It is my impression that some variation of this is at the root of most bad performance.
  • Make up a regular schedule for studying, and stick to it. If no homework or other work, go through your notes, solve proposed exercises. Count yourself lucky, I didn't have the Internet to trawl for material...
  • Leave a day off from studying each week (does wonders against ending all stressed out)
  • Don't study for the exam the day before. You may study other subjects, but not for tomorrow's exam (the last day you won't learn anything new, you'll only make yourself nervous).
  • Make sure you are well-rested for the exam (i.e., no allnighters the day before!)

With my TAs here we came up with the idea of proposing three problems for each weekly tutoring session: one solved by the TA (asking for input from the class), one to be solved by the class in open discussion (with hints/help from the TA), and one to be solved individually and handed in. Solutions to all are posted after the session. The problems handed in are checked just for rough relevance to the subject, not really graded, and their yes/no make up a 5% of the final grade. Seems to have made wonders in the one occasion we have used it up to here.

Give short homework, often. Keep them involved with the subject. If a week goes by without looking at the subject, reconnecting becomes very hard (even hopeless).

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That three problem session sounds interesting. How come it was only used once, if you don't mind me asking? – Carlos Apr 25 '14 at 2:49
@Carlos, just introduced the idea last term. – vonbrand Apr 25 '14 at 5:57

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