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55

I suspect that there are several different and interacting things going on here. It's likely that these students don't understand the "equals" sign in the same way that you do. An extensive body of research over several decades has documented that for many students the symbol $=$ does not indicate a relationship (that the thing on the left is identical in ...


37

When I'm the instructor for the course, I remind them of the time and location of my office hours by writing it on the board at the beginning of every lecture (if you're not the instructor, you can ask the instructor to do this). It sounds like overkill, but in my experience part of the reason students don't attend office hours is because it's just not a ...


33

The specific identity \begin{equation}\tag{A} \tfrac{1}{1 - \sin{x}} + \tfrac{1}{1 + \sin{x}} = 2\sec^{2}{x} \end{equation} as such is probably not often encountered, but simplifications akin to \begin{equation}\tag{B} \tfrac{1}{1-t} + \tfrac{1}{1 + t} = \tfrac{2}{1 - t^{2}} \end{equation} occur frequently. For example, integration via partial fractions ...


30

But, when teaching adults, I've found that I can't just tell them "this is the way it's done, get used to it." Good! Students (at any age) should never be satisfied with "This is the way it's done, get used to it", and teachers should never give that as an explanation. More than any other academic subject, everything in math should make sense. ...


26

It's perhaps worth noting that there are (at least) two distinct reasons why your students might offer no reply when you ask them if they understand what you've just explained: They might not understand what you've told them, or even understand enough to know what it is that they don't understand. This typically happens when the students lack (or have ...


25

Do NOT give exam questions that are intentionally more challenging than homework or in-class problems. I would recommend precisely the opposite. The point of the exam is really a spot-check that students know the basics and aren't just faking their way all through the class. If there is a time-limit, then that is already a lot more pressure/high-stakes ...


19

First: I do not think this is really an issue with a lack of understanding of the square root function. When someone writes $\sqrt9 = \sqrt3$ it means they are not thinking about what the equals sign even means. I have had a small amount of success with the following method, which relies on the students "believing in their calculator" as a source of truth. ...


17

When I teach courses, I usually see the majority of my students in office hours at least once a week. Here are some strategies for promoting office hours: Get a room. My office isn't nearly large enough to accommodate 10 students, and students who have to wait outside don't feel welcome. By reserving a classroom or other large room, I make sure there's ...


17

It's hard to admit you don't understand if you think you're the only one in the class that's lost (rarely actually the case, but individuals get self-conscious). I employed a system where if I wanted to check perceived understanding, I would do an "understanding check". Students would hold up 1-4 fingers: 1 - I'm so lost I can't even ask a good question. ...


16

I think a great reason to post homework solutions is so that students have a way of reviewing their homework before exams (it would be nice if they were to review homeworks whenever they get their graded work back, but this seems to be a rare practice). While a well-graded problem set should have comments pointing out any mistakes the student may have made, ...


16

Late answer, but since it comes up every semester I have a stock response that I'd like to share. The most common mistake in this vein is for a student to write $\sqrt{16} = \sqrt{4} = 2$ or $\sqrt{81} = \sqrt{9} = 3$. Frequently if I try to correct this they'll ask, "Where did the radical sign go?". When this happens, I go to the board and write something ...


16

As a TA, attend one of the class sessions and personally advertise your office hours. If the students see you in person, they may be more willing to come to you with questions. Also making yourself available outside of normal/regular office hours, giving them your email, and generally making yourself more approachable are all things you can do. I feel like ...


16

When I think of the what caused my students (or myself, back when I sat in lectures) to get to the point of an empty stare, it's that they were so confused that they didn't even know what to ask. They know they don't understand, but they lost the thread a while ago. So what can you do? Two things: Make sure students feel comfortable asking questions, so ...


16

The tutors need training. Any volunteer job has job requirements, and one of the requirements of this job [for it to be done well] is being able to tutor anything that comes along. The training can mainly involve intriguing problems that will help the tutors enjoy math themselves. Then, if they get stuck on a student's problem, they can seek help in a ...


15

I think at this level, and for a tutor (not a teacher), the main carrot (really more of a stick) is getting good grades, getting into a good college, etc. Most people really will never use this later, and for most others it's mostly a stepping stone to more advanced topics. As for how to present it to someone, try the handshake method (every term in one ...


15

I was always honest: You won't use this. If you don't like math by now, it's probably not going to be useful to you in your career or life, or ever again once you graduate. If you're like most people, you're going to completely forget this. That's not why we're doing it. Ever watch an athlete workout? Maybe do a bench press? Regardless of the sport he ...


15

I've created roughly 40 videos for an online course and a flipped course in biology, so I've got some experience making videos. I researched the maker of your videos, and he deliberately does NOT reveal his process. So I don't know if this is his method, but I'm comfortable that this would produce something very similar. Supplies: Computer (I use a ...


15

I think I need a simple graphic mnemonic he can use to remind himself which side the ones column goes on, and which direction we go in order to get bigger numbers. Suggestion: Start making routine use of the decimal point, and perhaps instruct the student to always write that as a guidepost; note that it always goes at the end of an integer (to the ...


15

Since you didn't mention what level your student is, I'll try to make my answer as general as possible, leaning more toward secondary math students. In my experience tutoring and teaching secondary level mathematics, I have observed that many students who struggle most have a number of skill domains that are far below grade level, and that these ...


15

How do I reach [this] kid? Let me be blunt: You probably don't. This is a person who is so intransigent that you effectively need to black-tag them. A hard lesson is that you can't save everyone. At this point the priority is to make some kind of defense so you aren't overly stressed, psychologically damaged, or burnt out. A couple personal reactions to ...


14

Definitions and other facts One thing I find particularly helpful with Linear Algebra is to help the student deal with the definitions in multiple ways. In Linear Algebra there are definitions, and there are properties that things have that are always true but aren't definitions. (They could have been chosen as definitions, but the chosen definition is more ...


14

Due to low enrollment, my AP Calc class was filled with the students who otherwise would have taken Pre-Calc this year. So you can imagine that "How much do you really need to know to see the bigger picture in calculus?" has been on my mind lately. Here's where my thoughts have fleshed out in regards to trig so far. Periodic behavior is widespread ...


13

I don't see my thoughts expressed by anyone here so perhaps I can chip in! I'm currently a graduate student but I was once that kind of bright bored student you are talking about. I had my fair share of boredom, lost motivation, exciting learning, disappointment and enthusiastic crankery. I can tell you that taking more interesting courses and finding good ...


12

One of the benefits of practice is a deeper understanding. I don't care how fast you get the answer, but I hope you are not adding by columns when you add 101+99. And that's why thinking of practice as equivalent to drill may be problematic. Maria Droujkova talks about how you need to love something to learn it well. If we want kids to learn the times ...


12

When I was a calculus TA last semester, I was able to recruit tons of students to my office hours. Here are some tips: Hold test reviews just before midterms/finals. These were my best attended hours! Also, some students who came to these started coming to other office hours later in the semester. At your test reviews, provide materials for the students to ...


12

As I imagine your interaction with the students, each of you had the following mental picture of what is going on. This is obviously a big assumption about the nature of the class and of your own personality, but the gap I describe is definitely there regardless of the details. You: A differential equation (which is synonymous with a system thereof) is a ...


12

The inabilities described in the question don't strike me as either particularly unusual nor as clear signs of a neurological learning disability. They strike me as more likely the passivity and mental shutdown that one frequently encounters in students who, whether consciously or unconsciously, are hiding a sense of incapacity, inability, confusion, etc. ...


11

Homework (and exam) solutions should be published, at the very least to enable students to request regrading. Yes, many students never look at the solutions (I even have a stack of homework and exams from last term that they didn't even bother to pick up). But that some don't bother isn't reason enough to deprive those who are interested of being able to ...


11

Nothing ironclad, but I have a couple of ideas. I hope they're of use. It's admirable that he has persistence in solving problems. This is, of course, an important mathematical practice identified by many (not to mention codified in CCSS.MATH.PRACTICE.MP1: "Make sense of problems and persevere in solving them." Most often teachers of mathematics must work ...


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