# HOW MANY MATH QUESTIONS IS ENOUGH?

#### Subhotosh Khan

##### Super Moderator
Staff member
I think the answer to the question "how much is enough" depends on the student's mental capacity/agility and physical capability - by a significant amount.

Initial investment of time and efforts are important - but "repetition" makes a student MASTER.

That is why - after 30 yrs. of working as a research engineer - I took up a teaching job at a community college. I need to be in fighting shape to tame those grandchildren and call me "grampy MASTER".

#### Dr.Peterson

##### Elite Member
1. The problems without answers does not serve a purpose. How would I know if my answers are correct or not?
As I've said (but keeping in mind that I have taught from these books, but not written or published them, so I can't be sure) the primary purpose of those problems is not for the self-learner, who can mostly ignore them, but for the teacher. I have often assigned a set of odd problems (not all of them, which would be an unnecessary burden for a good student) to be done to learn from (students who need more are encouraged to do more), and then a few even problems to be turned in and graded, as a reality check -- in effect, a take-home quiz. Too many students can think they have learned the material if they can eventually solve everything with the help of the answers in the back, but discover (if they even notice) that they can't do it without that crutch. Self-learners can use odd problems the same way, by not looking in the back until you would be ready to turn them in if you had a teacher. You can get your own reality check.

You could use the even problems the same way: After you think you've mastered a section, try some of the harder even problems to see if you are confident that you got it right (including checking your answers, an important skill that many students don't bother with because they can look in the back or have the computer check). If you aren't confident, go back and learn that material by rereading and doing more odd problems (which might include redoing those you had already done).

You could also ask here about those even problems you aren't sure of. (Don't bother asking about the odd ones, or the even ones you've checked and got right.)

And you generally don't miss anything by skipping the even problems; many books have problems in pairs, an odd and an even at the same level of difficulty, for the teacher to choose. (Some books don't, perhaps more often at higher levels. I have more than once found that I couldn't find an odd problem to teach a special case that is seen in a couple even problems.)

The important thing, as has been said, is to do as many as you need to feel confident that you could do the rest. That's entirely up to you, and entirely dependent on your own skills and weaknesses. You have to time to master the subject, if you work at it intelligently. Too many students in a class don't have (or take) that time, and don't work at it with that goal in mind, but just do the minimum they are required to do.

##### Junior Member
You must learn to check your answers. Everyone makes mistakes. Everyone must learn how to find and correct those mistakes. Learning from mistakes is very powerful.

I do not get the relevance. You can use paper and pencil without any need for LaTeX.

No. I do not understand. You have a certain limited amount of time when you are free and mentally capable of doing math. That I understand. Let’s say that is four hours. You need an hour to read a section carefully and thoroughly. That leaves only three hours to do problems. 180 minutes. With 24 problems, that is 7.5 minutes per problem. If you thoroughly understand the topic and work carefully, that seems ample: some easy problems will take far less, and the hardest problems may take ten or twelve minutes, but 7.5 minutes seems a generous average if you have thoroughly mastered the material.

Notice that four hours is the minimum you would expect to spend if you were taking a course. The idea that you can learn as fast when self-teaching as you can if taught by someone fully conversant with the material is simply ridiculous. You should expect that mastering calculus on your own will take at least two years. If you think that it can be mastered in three or four months by working only a few hours per week, you are wrong.

Continuity, Riemann sums, limits, etc. are the hardest parts of calculus. It took professional mathematicians two hundred years to develop these concepts. If it takes you six weeks to master them, that is hardly surprising. Most of calculus is considerably easier if your algebra is solid.
Understood. What about questions located in the CHALLENGING SECTION at the end of each chapter? These are questions that push the student's understanding of the chapter material. Most of the time, the questions involve some sort of proof. You say?

##### Junior Member
As I've said (but keeping in mind that I have taught from these books, but not written or published them, so I can't be sure) the primary purpose of those problems is not for the self-learner, who can mostly ignore them, but for the teacher. I have often assigned a set of odd problems (not all of them, which would be an unnecessary burden for a good student) to be done to learn from (students who need more are encouraged to do more), and then a few even problems to be turned in and graded, as a reality check -- in effect, a take-home quiz. Too many students can think they have learned the material if they can eventually solve everything with the help of the answers in the back, but discover (if they even notice) that they can't do it without that crutch. Self-learners can use odd problems the same way, by not looking in the back until you would be ready to turn them in if you had a teacher. You can get your own reality check.

You could use the even problems the same way: After you think you've mastered a section, try some of the harder even problems to see if you are confident that you got it right (including checking your answers, an important skill that many students don't bother with because they can look in the back or have the computer check). If you aren't confident, go back and learn that material by rereading and doing more odd problems (which might include redoing those you had already done).

You could also ask here about those even problems you aren't sure of. (Don't bother asking about the odd ones, or the even ones you've checked and got right.)

And you generally don't miss anything by skipping the even problems; many books have problems in pairs, an odd and an even at the same level of difficulty, for the teacher to choose. (Some books don't, perhaps more often at higher levels. I have more than once found that I couldn't find an odd problem to teach a special case that is seen in a couple even problems.)

The important thing, as has been said, is to do as many as you need to feel confident that you could do the rest. That's entirely up to you, and entirely dependent on your own skills and weaknesses. You have to time to master the subject, if you work at it intelligently. Too many students in a class don't have (or take) that time, and don't work at it with that goal in mind, but just do the minimum they are required to do.
1. I always do my best but better than my best is insanely impossible.

2. I don't have time to read the textbook lessons. I usually make use of the chapter outline as my guide. For example, Section 1.5 is all about Limits at Infinity. I then search You Tube for Limits at Infinity video lessons. I take notes on everything said in the video lesson. I work out all sample questions with the video instructor. Is this a good way to learn the material?

3. We are making this odd/even number questions harder than it is. Odd numbers have the answers in the back of the book.

4. Checking for myself to see if the answers are right is ok when questions easy.

5. Just because I am studying Calculus 1 does not mean that Precalculus should be ignored. You say?

#### LCKurtz

##### Full Member
Just curious nycmathdad, have you in the past gone by the name "Harpazo" on another site?

##### Junior Member
Excuse me if I misinterpreted your original post.

You said that a section has about twenty-five exercises and that you spend a week going through a section. I was assuming that you spent about an hour a day. That adds up to 7 hours to grasp the concepts and do the exercises.

A section has more than 70 questions. I select 25 per section. Well, I used to answer 25 questions per section. The goal was to answer 20/25 correctly or 80.percent. Less than 80 percent meant a repetition of the section is needed.

Now, you may average less time per week. Perhaps you spend two hours a week on average. Notice that that is less than half the time you would spend if you were taking a class and doing homework. If that is all the time you can afford, then that is all the time you can afford, but it certainly is not excessive for self-learning.

I am doing the best I can under the circumstances. I can't do more than my best.

My fundamental point is that the more problems you do related to a particular topic, the more likely you are to grasp and retain the mechanics and applicability of that topic. If you can solve correctly 25 problems in 25 minutes, you definitely have that topic. On the other hand, if it takes you twenty minutes of careful work for you to get the wrong answer on a single problem, it makes no sense to start on another problem immediately; it makes more sense to go back to studying the text or asking a question here than continuing with the problems because that much work to get a wrong answer means you are missing some vital piece of what the section is about.

I have no time to read the lesson material per section. I utilize You Tube video lessons for learning. I take notes when watching video lessons and then answer questions.

As a self-learner, the problems are the way you confirm that you understand; they are the only test a self-learner gets.
Keep in mind that this is just for fun. I am never going to be a math teacher. Those days are behind me. There was a time when aiming to teach as a full-time job was my desire. I quickly got a reality check when I found out just how difficult the journey to become a certified teacher really is.

##### Junior Member
Just curious nycmathdad, have you in the past gone by the name "Harpazo" on another site?

#### jonah2.0

##### Full Member
Beer soaked comment follows.
Just curious nycmathdad, have you in the past gone by the name "Harpazo" on another site?
He is Harpazo.

#### JeffM

##### Elite Member
1. I always do my best but better than my best is insanely impossible.

2. I don't have time to read the textbook lessons. I usually make use of the chapter outline as my guide. For example, Section 1.5 is all about Limits at Infinity. I then search You Tube for Limits at Infinity video lessons. I take notes on everything said in the video lesson. I work out all sample questions with the video instructor. Is this a good way to learn the material?

3. We are making this odd/even number questions harder than it is. Odd numbers have the answers in the back of the book.

4. Checking for myself to see if the answers are right is ok when questions easy.

5. Just because I am studying Calculus 1 does not mean that Precalculus should be ignored. You say?
You have a text, but you do not have time to read it. Nor do you have time to do problems.

Let’s be honest: you do not have enough time available to learn calculus.

#### jonah2.0

##### Full Member
Beer comment follows.
... 2. I don't have time to read the textbook lessons.
I usually make use of the chapter outline as my guide.
And that explains almost everything.
All war and peace posts of good advices and suggestions thrown out the window.
Old habits die hard. Still the same Harpazo ignoring excellent tactics and strategies for studying effectively.
For example, Section 1.5 is all about Limits at Infinity. I then search You Tube for Limits at Infinity video lessons. I take notes on everything said in the video lesson. I work out all sample questions with the video instructor. Is this a good way to learn the material? ...
You "don't have time to read the textbook lessons" but you have time to "take notes on everything said in the video lesson"???
I say read the textbook lessons first thoroughly and carefully and supplement it with video lessons on the parts that are not clear to you.

#### LCKurtz

##### Full Member
Because, given that you are Harpazo, I have seen enough to know not to waste time giving you advice.

##### Junior Member
Because, given that you are Harpazo, I have seen enough to know not to waste time giving you advice.
This is enough to place you on the Ignore List. Congratulations!

##### Junior Member
You have a text, but you do not have time to read it. Nor do you have time to do problems.

Let’s be honest: you do not have enough time available to learn calculus.
Did you not read my reply? I take notes watching video lessons from instructors. I study the notes before answering questions.

#### JeffM

##### Elite Member
Did you not read my reply? I take notes watching video lessons from instructors. I study the notes before answering questions.
Different teachers have different approaches. The problems that accompany a text are almost invariably associated with that text rather than with a lecture given by someone different from the text’s author. Furthermore, your notes can never be as carefully thought through as a text that is revised, edited, and peer reviewed.

There is another student here who jumps from source to source. He can never figure out why he cannot make sense of different voices, each explaining things in different ways, and wastes his time trying to reconcile different approaches rather than trying to grasp the meaning behind a single approach.

Finally, a carefully written text can be read in a quarter of the time it takes to listen to a lecture. If time is scant, learning by lecture is amazingly inefficient.

The method you have chosen would not be what I would advise, but you are a grown man. You need to find what works for you given your time constraints.

#### jonah2.0

##### Full Member
Beer soaked ramblings follow.
Because, given that you are Harpazo, I have seen enough to know not to waste time giving you advice.
This is enough to place you on the Ignore List. Congratulations!
How much is enough?
Do you follow a formula on when to place members on the "Ignore List"?
Must they earn a certain minimum of something?

##### Junior Member
Different teachers have different approaches. The problems that accompany a text are almost invariably associated with that text rather than with a lecture given by someone different from the text’s author. Furthermore, your notes can never be as carefully thought through as a text that is revised, edited, and peer reviewed.

There is another student here who jumps from source to source. He can never figure out why he cannot make sense of different voices, each explaining things in different ways, and wastes his time trying to reconcile different approaches rather than trying to grasp the meaning behind a single approach.

Finally, a carefully written text can be read in a quarter of the time it takes to listen to a lecture. If time is scant, learning by lecture is amazingly inefficient.

The method you have chosen would not be what I would advise, but you are a grown man. You need to find what works for you given your time constraints.
I agree with everything you said. However, textbook lessons can look like Chinese backwards depending on the topic. Can you imagine what a Calculus 3 textbook must be like? I can read a Calculus 3 lesson on double and triple integrals and not understand a single sentence. The video lectures remove all the math jargon that students fear and hate. I'm not talking about prealgegra topics. I am trying to learn Calculus 1 to a decent level. Textbook lessons are a bit fuzzy for me right now.

#### jonah2.0

##### Full Member
Beer soaked ramblings follow.
Translation: I hear you but I prefer my system.

#### JeffM

##### Elite Member
I agree with everything you said. However, textbook lessons can look like Chinese backwards depending on the topic. Can you imagine what a Calculus 3 textbook must be like? I can read a Calculus 3 lesson on double and triple integrals and not understand a single sentence. The video lectures remove all the math jargon that students fear and hate. I'm not talking about prealgegra topics. I am trying to learn Calculus 1 to a decent level. Textbook lessons are a bit fuzzy for me right now.
The “math jargon” is what is exact. Any other words may seem familiar and thus easy to understand, but they do not describe things precisely, and thus do not really explain anything.

$$\displaystyle f(x) \text { is continuous in } (a, \ c) \iff a < b < c \implies f(b) \text { is a real number, and} \lim_{x \rightarrow b} f(x) = f(b).$$
Now to truly understand that you also need to understand

$$\displaystyle \lim_{x \rightarrow p} f(x) = L \iff \text {for arbitrary } \epsilon > 0, \ \exists \ \delta > 0 \text { such that } 0 < |x - p| < \delta \implies | f(x) - L | < \epsilon.$$

Saying that you do not want to grasp those symbols and definitions is effectively saying you do not want to understand what calculus is about.

Grasping limits and continuity and differentiability means looking at examples that fit and examples that do not fit those definitions until you yourself can determine what functions are continuous or differentiable. Lectures do not give you facility with concepts. Repetitive application does.

Math is a language. Saying you do not want to learn its “jargon” is like saying you want to learn French without learning French words. You won’t learn French from listening to lectures in English.

Why do you study limits?

Because limits are central to the definitions of continuous and differentiable functions.

If a function is not continuous on an interval, it is not differentiable on that interval.

If a function is not differentiable on an interval, differential calculus does not apply to that interval.

Newton, who was certainly a great mathematician, had trouble understanding the text of Descartes. Whenever it grew fuzzy, he went back to the beginning and started over. Reading math is not like reading a novel.

##### Junior Member
The “math jargon” is what is exact. Any other words may seem familiar and thus easy to understand, but they do not describe things precisely, and thus do not really explain anything.

$$\displaystyle f(x) \text { is continuous in } (a, \ c) \iff a < b < c \implies f(b) \text { is a real number, and} \lim_{x \rightarrow b} f(x) = f(b).$$
Now to truly understand that you also need to understand

$$\displaystyle \lim_{x \rightarrow p} f(x) = L \iff \text {for arbitrary } \epsilon > 0, \ \exists \ \delta > 0 \text { such that } 0 < |x - p| < \delta \implies | f(x) - L | < \epsilon.$$

Saying that you do not want to grasp those symbols and definitions is effectively saying you do not want to understand what calculus is about.

Grasping limits and continuity and differentiability means looking at examples that fit and examples that do not fit those definitions until you yourself can determine what functions are continuous or differentiable. Lectures do not give you facility with concepts. Repetitive application does.

Math is a language. Saying you do not want to learn its “jargon” is like saying you want to learn French without learning French words. You won’t learn French from listening to lectures in English.

Why do you study limits?

Because limits are central to the definitions of continuous and differentiable functions.

If a function is not continuous on an interval, it is not differentiable on that interval.

If a function is not differentiable on an interval, differential calculus does not apply to that interval.

Newton, who was certainly a great mathematician, had trouble understanding the text of Descartes. Whenever it grew fuzzy, he went back to the beginning and started over. Reading math is not like reading a novel.
1. You are wrong by saying that I have no real desire to learn calculus.

2. I do not disagree with what you said about MATH JARGON.

3. Jeff, I do not aim to become a calculus champion. I just want to learn enough to answer regular calculus questions taught in high schools and colleges across the globe.

4. I am not going to ever teach math for a living. I am 55 years old. It's too late for me.

5. If calculus 1 becomes a nightmare, no problem. In that case, I simply will return to precalculus grounds and "master" that course, if possible.

Questions:

A. I am thinking about revisiting precalculus every now and then. You say? I can do occasional precalculus questions and calculus 1 at the same time.

B. I just finished Section 1.2, which covers Properties of Limits. The section is not too bad. However, I came across some challenging questions at the end of Section 1.2. I posted new threads in the Calculus forum. Can you help me with 110 and 111?