How can math teachers effectively engage students during lectures?

[kdinser]

I'm currently taking a first course in differential equations and, just like last semester, finding 75% of the 4 hour lecture to be worthless to me.

The professor walks in and asks about homework problems and works through any that people had trouble with. I generally find this useful, it's interesting to see how he does some of the same problems I worked on.

Then he starts going over new material. Seeing him go over the theory is a good thing, the book usually doesn't give enough detail to get the whole picture.

Then the trouble starts. After teaching a new concept, he goes step by step through 3 to 4 examples, taking about 15 minutes per problem. This it actual torture to me. I'm not good at sitting, I built a giant dry erase board in my room so I could move around while doing my homework. Personally, I don't get a thing out of watching someone solve math problems and I don't see what anyone else could get out of it. It's a new concept, they are probably going to be 2 to 3 steps ahead of me at any given time anyways.

If the point is to give us a few fully worked problems to help us understand the concepts and homework, then DO THEM AHEAD OF TIME AND JUST HAND THEM OUT! We could cover twice the material that way. This is a mid level math class. If you have gone past calc I, it's a good bet that mathematics is going to be a big part in your chosen profession. Shouldn't the hand holding be about over by now? Shouldn't the "I suck at math" and "I hate math" people have left the building long ago?

Every math teacher I've ever had stresses, at the start of the semester, how important doing the homework is. You can only learn math by practicing it, repetition is the key, you can only learn to solve problems by solving problems, ect... For the most part, I agree, I learn a new concept, then I practice it until I feel I get it.

Oh well, I think I had a question in there at some point, Oh yea, is this what I'm doomed to for the rest of my math courses? I'm planning on getting a math minor.

 

[franznietzsche]

If the point is to give us a few fully worked problems to help us understand the concepts and homework, then DO THEM AHEAD OF TIME AND JUST HAND THEM OUT! We could cover twice the material that way. This is a mid level math class. If you have gone past calc I, it's a good bet that mathematics is going to be a big part in your chosen profession. Shouldn't the hand holding be about over by now? Shouldn't the "I suck at math" and "I hate math" people have left the building long ago?

They're called "engineers". They're everywhere.

 

[Moonbear]

There are people with many different learning styles out there, and professors have to teach to all of them. For some, it will help solidify the theory to see it applied in a few example problems right away since it might be a few hours before they can sit down to homework and check their understanding of the lecture material. There are probably quite a few in class who find going over the homework problems to be a waste of time, especially if they got the solutions okay (that used to be the most painful part of math classes for me, going over all the homework problems that I already knew how to solve).

Though I'd get super-fidgety in a 4 hour class too! It's probably more that you're getting tired by the end and not getting much out of it because it's running so long that's more of a problem than that the examples aren't good to see. I've never had to sit through a lecture longer than 3 hours, only labs were longer because you needed the time and you didn't have to sit still. I had a few graduate level classes that would run 4 hours, but those included a good deal of discussion more than lecture, so you didn't really notice how long it was as long as we took a break halfway to use the restrooms.

It seems the problem isn't so much how your prof is teaching so much as whoever was in charge of scheduling the class really would have served you all better by giving you more shorter lectures instead, or dividing the class into a lecture session to go over theory and examples and a recitation session to go over homework assignments and problems.

 

[ktpr2]

"then DO THEM AHEAD OF TIME AND JUST HAND THEM OUT! We could cover twice the material that way."

Right, but you're assuming you can absorb and comphrend twice the amount of new material. Most people can't and hence the structure of the class. What would you propose the teacher do with the rest of the class if he didn't go over problems?

 

[kdinser]

Right, but you're assuming you can absorb and compared twice the amount of new material. Most people can't and hence the structure of the class. What would you propose the teacher do with the rest of the class if he didn't go over problems?

How about this, teach a new concept and do one example, then hand out 2 or 3 problems for us to try. This can be done individually or in small groups. Give us the same amount of time that the professor would have spent doing them. Allow a few mins for questions. At the end of class, hand out solution sheets with the problems fully worked so we can compare our work to the professors. At least this would give us some of our own work to take home and study from.

Moonbear, yea, a 4 hour lecture class is just wrong . Unfortunately, I'm finishing up my math and physics requirements at a junior college. Most of the students that are taking anything past calc I are adult students with full time jobs. Calc II was only offered last semester one day a week, Tuesday night 5:30 to 9:30. I didn't think it could get any worse then that, but diff eq is also only one day a week, 6:00 to 10:00 :yuck: :zzz:. I'm taking calc III in the spring, it's only offered 2 days a week, 4 hours per day. I think I'll just go back to my previous 4 year college to take that one. Their 4 hour spring/summer classes were 2 hours per day, 4 days a week.

 

[Reason]

"then DO THEM AHEAD OF TIME AND JUST HAND THEM OUT! We could cover twice the material that way."

Right, but you're assuming you can absorb and comphrend twice the amount of new material. Most people can't and hence the structure of the class. What would you propose the teacher do with the rest of the class if he didn't go over problems?

If you hand the solved problems out, at least the students don't wast their time by writing down the solution. Instead they could try to understand it already. Because believe me, when you are writing things down you don't always have the time to understand things too. But this doesn't apply to math alone. Almost all scientific courses suffer from this problem.

 

[arildno]

I never found writing down productive.
Rather, I paid attention to the new concepts learned, and tried to anticipate how the professor would implement them in the worked examples. In this manner, I would find out whether I understood the concepts well enough to build further upon, or if I needed to self-study those concepts outside the lecture hours.

 

[brewnog]

I'm not good at sitting, I built a giant dry erase board in my room so I could move around while doing my homework.

I think this is great, wish I'd thought of that.

Well you've realized that you learn much better this way, that's probably the hardest step. (Admitting you have a problem is the hardest part...! ) I know how you feel, I used to walk out of my maths lectures when the theory stopped and the dreaded 'worked examples' began.

 

[BobG]

The professor walks in and asks about homework problems and works through any that people had trouble with. I generally find this useful, it's interesting to see how he does some of the same problems I worked on.

Then he starts going over new material. Seeing him go over the theory is a good thing, the book usually doesn't give enough detail to get the whole picture.

Then the trouble starts. After teaching a new concept, he goes step by step through 3 to 4 examples, taking about 15 minutes per problem. This it actual torture to me. I'm not good at sitting, I built a giant dry erase board in my room so I could move around while doing my homework. Personally, I don't get a thing out of watching someone solve math problems and I don't see what anyone else could get out of it. It's a new concept, they are probably going to be 2 to 3 steps ahead of me at any given time anyways.

If the point is to give us a few fully worked problems to help us understand the concepts and homework, then DO THEM AHEAD OF TIME AND JUST HAND THEM OUT! We could cover twice the material that way. This is a mid level math class. If you have gone past calc I, it's a good bet that mathematics is going to be a big part in your chosen profession. Shouldn't the hand holding be about over by now? Shouldn't the "I suck at math" and "I hate math" people have left the building long ago?

Every math teacher I've ever had stresses, at the start of the semester, how important doing the homework is. You can only learn math by practicing it, repetition is the key, you can only learn to solve problems by solving problems, ect... For the most part, I agree, I learn a new concept, then I practice it until I feel I get it.

It's hard to say doing 3 or 4 examples is excessive. It depends on the examples he chooses. If there's 3 or 4 different situations you can apply a concept to, then he should demonstrate each one. Integration by parts is a good example - he should do an example using a product of two functions, a natural log, and a trig function with no quick antiderivative, cyclic functions, (any others?).

Once you start the problem, it winds up being the almost the exact same thing over and over , but it wouldn't be obvious that that was the right technique to use until it was demonstrated.

Ideally, he would start getting more student participation to solve the parts that are just a repetition of the previous example. Or, better yet, once he's started the problem, let the students finish it. How he handles that kind of thing depends on instructor style (some 'styles' are definitely better than others, even if quite a few different ones at least meet the minimum).

Edit: I definitely agree with the 'hand holding' part. Instructors shouldn't hold hands with students :rofl: . If the student isn't taking any responsibility for learning the material, why should the teacher care (except for the fact that it's his job, like it or not).

The students sure don't put up with that (at least none of the ones that have had jobs). Just watch how the groups form when group projects are assigned. All the folks who are still doing their homework at the start of class, or never have time for their homework at all, wind up sitting at their desks wondering why they're not part of a group until the teacher finally steps in - and then cruelly puts all the leftovers into one group. Just concientously preparing them for their future life in the workplace, I'm sure. (Group behavior is so interesting - almost as interesting as the book "Lord of the Flies")

 

[ktpr2]

"The students sure don't put up with that (at least none of the ones that have had jobs). Just watch how the groups form when group projects are assigned."

I find just the opposite occurs: the people who don't do homework or struggle with it instantly group up and the brainiacs are all alone and/or group up with other brainiacs.

Also, I've never found repetitious examples boring, at least when the teacher asked the students what to do next or what's the result from a sequence of work. If it's boring, then try to work ahead of the teacher and figure it out in your head before he does. At least when he calls on people you can always be the first one to answer.

 

[Moonbear]

The students sure don't put up with that (at least none of the ones that have had jobs). Just watch how the groups form when group projects are assigned. All the folks who are still doing their homework at the start of class, or never have time for their homework at all, wind up sitting at their desks wondering why they're not part of a group until the teacher finally steps in - and then cruelly puts all the leftovers into one group. Just concientously preparing them for their future life in the workplace, I'm sure. (Group behavior is so interesting - almost as interesting as the book "Lord of the Flies")

Oh, watching groups work together is really fun. I often see a group of slackers do their best to coalesce around the one smart kid in the hopes the smart kid will do all the work for them, and that's usually exactly what happens, which really defeats teamwork. The funny thing is if you ask them to keep a journal of their activities on the team, it's so blatantly obvious who is doing all the work and who is just trying to get the credit for standing around doing nothing. On the other hand, if you can intervene and get one slacker onto a team of smart kids, the slacker might actually pick up a few good study habits.

The problem I have with the way most team assignments are done is that nobody ever takes the time to teach the students how to be an effective team, they expect them to somehow naturally know how to do this. The greatest resistance to team work often comes from the smartest kids because they are used to working independently and not needing to ask for help, so wind up feeling like they are going to just be carrying the team rather than putting effort into using their knowledge to further build up their team.

The greatest success I've ever seen with teamwork was to get together a group of students who all got C's on their first biology exam who were mostly pre-med, so scared to death of getting a C in the course, thus motivated to do something to correct it. There were about a dozen of them who I met with for an hour every week and rather than tutoring, I facilitated their study group to help keep them on track and teaching them what questions they should ask each other and really only helping when they got stuck. By then end of the course, they were the A students. They stuck together as a team through their other major courses and wound up at the top of their class (and I was really happy when a few "saw the light" and decided to head into research instead of medicine). I wish I had the time to do that with more than just a handful of students. Then again, they were the only ones who showed up when I offered the assistance, so it might not have worked as well with those who didn't already recognize a need to do something different in their studying.

 

[derekmohammed]

Personally I find it ok when an instructor does a few more example problems then he should. This usually shows that the prof accually cares about his class; since ussually the problems that he spends extra time stressing the meathod almost more times then not show up on the exam!
This is a huge advantage.
I have to agree with bobG that it is the begininng of a problem that causes the trouble, that is why examples show us the tricks to easier solutions on the exam. Also the big advantage of some examples is that it cuts down on your study time since you have a clear definitoin of what the applications are and what diffrent ways the problem could be asked.
Thats just my opinion...

PS: Althought it does get very boring sometimes we must remind our sometime overinflated egos that it is helping us not hurting us. (I have been known to sometimes fall asleep too... :tongue: )

Derek

 

[matt grime]

If you don't do examples you get far more complaints than if you do. And as complaints are what matters in teaching evaluations (or rather the lack of them) then this is what will happen. Sorry, but the majority of students see evals as a good chance to insult someone anonymously, though they may think they are being productive. Departments give them a lot of weight. Consequently damage limitation goes on.

One more thing. It's a good idea for students to see how to write maths properly since in my experience writing a clear exposition of an answer without undefined meaningless symbols, with words that explain the steps, and that are even logically sound is one of the hardest things to master.

 

[arildno]

One more thing. It's a good idea for students to see how to write maths properly since in my experience writing a clear exposition of an answer without undefined meaningless symbols, with words that explain the steps, and that are even logically sound is one of the hardest things to master.

I second this:
Too many students become confused by their own sloppy notation as it is.
To be clear about the assumptions (for example, by writing them down), and using an unambiguous notation, is actually a skill which needs to be developed (much like arithmetical manipulation is an elementary skill one must develop).

To see how a problem ought to be done, is very important, IMO.

 

[Gokul43201]

One more thing. It's a good idea for students to see how to write maths properly since in my experience writing a clear exposition of an answer without undefined meaningless symbols, with words that explain the steps, and that are even logically sound is one of the hardest things to master.

But this should be covered in lower level courses. And I'm not sure that repeatedly working out problems in front of the class helps this cause any.

I tend to agree with the OP in that for the most part, it is a colossal waste of time to have the teacher solving problems for a good part of an hour. Handing out solved problems sounds very reasonable to me. Also, I think having a 4 hour lecture is terribly inefficient.

 

[matt grime]

Depends on what level of course you're thinking about, I suppose.

A 4 hour lecture does indeed sound unreasonable though.

 

[mathwonk]

the point is to realize you are involved with other people in a group effort to learn. it helps if you care about whether the other people learn too, who have different learning styles from you. You seem to be unable to stomach any class unless it is oriented primarily to you and your speed of learning.

In that case, instead of fuming because the rest of the world does not change to suit you, you should be more careful to seek out a class where the average student resembles yourself. Either find a more advanced course, or a more advanced school, or else tell the professor you are in need of more challenging work, or else take the initiative to seek it out and do it on your own.


I have had classes in which I went very slowly, and yet the advanced students in the class took pleasure in being able to follow everything, and make helpful suggestions to aid the others who were struggling. Then I have had classes where the few students who knew more background than others were angry because the class went slowly enough to fill in background that most others needed. eventually those more advanced students quit working because they were so unhappy, and ultimately learned very little.


We are all in this boat together. When you drive home, do you blow the horn because the lug in front does not get out of your way? or pray to God that we all arrive safe?

We have a choice as to the world we personally live in. peace.

 

[kdinser]

Bottom line is, a 4 hour lecture on any subject is torture to me. I wish I would have looked at another local school in my area for my math classes. If it was 2 days per week at 2 hours per lecture, I doubt I would have had problem putting up with the pace of the class.

I wouldn't say that I want the class geared towards my learning style, it just seems to me that many classes are geared around people that want to be held by the hand. Frankly, lectures should be a place to learn concepts, not master them. I consider myself to be of average intelligence, if I can get it in one example, odds are, most others will also. If not, there are a lot of resources out there. Your not supposed to be able to master the material from a few examples in class. You have to actually do a few of them yourself to be ready for a test. If people need more examples to understand a concept, that's what office hours are for, or TA's, or tutors, or books, or websites, or this forum, or fellow students, or your parents... and so on. Again, my econ teacher had a great quote he used to use, "I can not teach you the material, I can only facilitate the learning process". I wish he was teaching my diff eq class.

 

[ktpr2]

It's life. How you adapt and thrive determine how well you do. I'm making lemonade :D