The other day a student came to visit my office hours for some homework help. At the end of the visit, I wished her good luck with the course. She mentioned that the class (beginning calculus) was going to take a lot of hard work and studying. I didn't say anything to disagree with her, and I let her go on her way*. I do, however, believe firmly in studying smarter, not harder, and that it doesn't take "a lot of work" to do well in a math class.
My Personal Plan
When I take a traditional math class (i.e. lectures, homework or quizzes, and exams), I do all of the homework by myself. As I mentioned here as well, the struggle of doing this does more to cement things in my mind than anything else I could do.
How do I do a homework assignment that I don't already know how to do? How do I figure it out?
Here's my plan of attack.
Before Working on Problems
I don't do anything before working on problems. I start by working on problems. Some people may start by trying to learn the material needed for doing a problem, and then they do the problems. I don't like that approach. To me, it's hard to learn math until I see how I can use it in a problem. Therefore, I dive right into the problems and then go back later to see what I need to know.
How I Do a Math Problem
Here's my plan of attack.
Before Working on Problems
I don't do anything before working on problems. I start by working on problems. Some people may start by trying to learn the material needed for doing a problem, and then they do the problems. I don't like that approach. To me, it's hard to learn math until I see how I can use it in a problem. Therefore, I dive right into the problems and then go back later to see what I need to know.
How I Do a Math Problem
- After reading the problem, make sure I know what the problem is saying. On a simple problem this may be easy, but on a more complex problem it may take time just to understand the problem enough to proceed with it.
- Once I understand what I need to figure out, I will be somewhere between knowing exactly what to do to solve the problem and having no clue whatsoever. In the case that I have no clue, I look back at the math book. In most any math class the material in the section being study is the key information needed to solve a problem. I especially key in on examples given in the explanation portion. Even up to beginning calculus, a lot of problems can be solved by just copying the steps of an example**.
- If there are any more holes in my knowledge, I will look at either other sections of the book or on the Internet (Wikipedia is great for this).
- If possible, I check my answer. If I am incorrect, I go back through the problem again and carefully analyze what I am doing to see what I have done wrong. If I simply made a careless error, I will probably catch it by doing this. If I have a larger gap in understanding, I will have to basically re-do steps 1-3.
If I diligently and patiently follow the above steps, I do all of my homework problems by myself. If a problem is hard, I often repeat these steps in between doing a lot of thinking.
Follow Up
There are a few natural ways I follow up on what I have learned from a homework assignment. First is going to class. Seeing the teacher use the material I have worked on can really help make that material be even clearer for me. Watching someone who knows what they're doing do math for some reason conveys more information than just reading it in a book.
Another thing is to see homework solutions. For a problem I did correctly, this can confirm my knowledge of it or even show me an easier way to do things. For a problem I did incorrectly, this can help show me what I need to do later. In the case that I did something incorrectly, unless I made a small, careless error, what I ought to do to is do more problems like the one I missed; just knowing about how to do something is a lot less effective than doing it and applying it myself.
Follow Up
There are a few natural ways I follow up on what I have learned from a homework assignment. First is going to class. Seeing the teacher use the material I have worked on can really help make that material be even clearer for me. Watching someone who knows what they're doing do math for some reason conveys more information than just reading it in a book.
Another thing is to see homework solutions. For a problem I did correctly, this can confirm my knowledge of it or even show me an easier way to do things. For a problem I did incorrectly, this can help show me what I need to do later. In the case that I did something incorrectly, unless I made a small, careless error, what I ought to do to is do more problems like the one I missed; just knowing about how to do something is a lot less effective than doing it and applying it myself.
What Kind of Skills Does this Require?
Being able to do this doesn't require extensive knowledge of a subject. There are a few skills which get used, however.
Being able to do this doesn't require extensive knowledge of a subject. There are a few skills which get used, however.
- Figuring out what a problem is saying is a skill. I have a hard time describing how to do that because what I basically do is a lot of visualization and other mental gymnastics.
- In looking at the book and looking at the examples, I need to be able to compare what's there to the problem I am working on. Once I have found a match, I need to have the skill of taking the example and picking out the essential steps which I then have to transfer to my problem. Basically, this skill is recognizing and reproducing patterns.
- Doing a problem itself relies on a lot of math skills. For example, a calculus problem could involve techniques from high school algebra, geometry, trigonometry, and pre-calculus, sometimes all in the same problem.
My opinion is that these skills are developed by doing. In my mind it's like learning to play a piano: I can't just read a book or have someone explain to me how to play a piano, I need to put in a lot of practice time actually playing the piano. In my opinion, therefore, if someone is deficient in these areas, they can improve only through a lot of practice doing problems.
The Benefits
In my experience, doing this means I don't have to study that much for a math class. Once I have done my homework, I have learned the material. Until I have done my homework, I don't really know the material that well. That's what it comes down to. (I just wish some other people got that too.) If I did my homework correctly, the only extra studying that I end up doing is the short time I spend refreshing my memory of something I learned so I can take a test. If I didn't do my homework 100% correctly, I figure out what I did wrong and then put in a modest amount of time on practice problems***. Either way, I don't end up investing that much time into a math class. (talk about testing)
To someone who may not be "good" at math, the benefits to focusing on doing problems is the same.
Notes:
* = I didn't argue with her at all because, among other things, she wouldn't have believed me.
** = If this is the case, however, then the problem becomes too easy. As a way of cementing my knowledge, I will try to do subsequent, similar problems from memory.
*** = To get practice problems, I would look to a few sources. I would do similar problems from my math book for the class. I would look on the course website for old midterm exams. I would google the topic I am having trouble with to see what I could get.
*** = To get practice problems, I would look to a few sources. I would do similar problems from my math book for the class. I would look on the course website for old midterm exams. I would google the topic I am having trouble with to see what I could get.
