Enjoyment and Interest
When a problem is too easy, it is not interesting or enjoyable. It is boring. If your student enjoys a problem, your student is probably learning something, although it might be solidifying a skill rather than developing some new insight.
If a problem is too difficult, it will probably not be interesting, enjoyable, or a learning experience. No harm is usually done.
So in some sense, the problem is just finding interesting and enjoyable problems. If you do that, you have succeeded.
Pre-Schoolers and Very Young Children
Your very young child is not a captive audience. You are stuck trying to find problems your child enjoys, which is the right thing anyway. Then it is a matter of finding a time when your child is interested. I attacked the boring times of being in the car or sitting in a restaurant. Sometimes my daughter wanted to do stick problems; sometimes she didn't. So if your goal is to produce a precocious math genius, it probably won't work. Try to produce a happy child.
The ideal situation is to begin with a young child who has not learned to hate math or to memorize. The problem then is even beginning to imagine how to teach those students.
This is the situation that most of my problem sets are built for. First of all, believe it or not, 4 + 3 = 7 isn't interesting for very long. That works for 4 or 5 year old. Believe it or not, you can give interesting conceptual problems to young children. You just have to start easy. For example, I started my stick problems with my daughter when she was 5. I started computer programming problems with 8-year olds.
Currently, I am in this exact situation. So I am developing problem sets. They will be added to the website. Hopefully, they can kick start the process. In other words, they can teach the fundamentals, so that you can then just give your student one of the many problems that are lying around.
I should note that in a carefully constructed problem set, the student is never faced with the difficulty of making two conceptual leaps at the same time. I should also note that one of the basic skills of math problem solving is taking a problem you can't solve and trying to simplify it so that you can solve the simpler problem. The well-constructed set then does not teach this skill. I am not trying to criticize the well-constructed set, I am trying to point out that it is not necessary or complete.
By college, or maybe even high school, a student has been taught a lot of math. The problem is that the student doesn't remember the math. Worse, the student does not like math, because it has been taught as memorization. Even worse, the student might not know how to solve a problem except by being taught a process to memorize.
I actually tried to teach bright math-incompetent college students once. A few students volunteered. The problem of 3 numbers that add to 100 is a good start. They have not been taught a process for solving it, and it starts them out thinking. It then neatly segues into finding their pattern, which then segues into using the pattern. This teaches the basic skill of connecting thinking to the manipulation of numbers on a paper. Most students mindlessly manipulate numbers on a paper with no understand of what they are doing and no connection to what they are thinking.
My next step was to give problems. Since they were all studying for the math test in the GRE exam, it was no trouble to find problems. After they had solved their problems, it was useful to have them each present their solution. They usually had different solutions, and the different solutions were interesting. Remember, my goal wasn't to teach a procedure for solving a problem, it was that they be able to solve problems. Each different method used a different tool that could be useful elsewhere.
Yes, You Can See Learning
Your student is trying to solve a problem. All of sudden, there is a smile. Eyes light up, and the student momentarily relaxes.
Learning mental models is difficult. Your student has a problem that he or she does not know how to solve. Then, when insight occurs, when the pieces of the brain have rearranged into a new or better mental model, it is enjoyable. So if your student is both working hard and enjoying himself/herself, you can be assured that learning has occurred.
All the available research, and my experience, suggests that students do not forget this type of learning.
Incompetents, Competents, and Lovers
Most people in our society are math incompetent. If you are a parent wanting to teach math to your child, you are probably math incompetent. Math incompetents were taught by the standard method of memorizing procedures. They do not understand math. They do not enjoy math. This is not their fault. It is the fault of how they were taught. It is not their teachers' fault, because their teachers did not know better. (But now you know better.)
Some people are math lovers. I am one. These people love doing math. No, they do not enjoy learning that 4 + 3 = 7. That's not math. That's memorization (forming a habit really). They enjoy math problem solving. Or they will enjoy games, or computers, or machines. They will find some outlet for their math ability, or they will not be happy. If you are a math phobic parent with a math loving child, you can and should use the problems-first method. Your child will love it. Yes, you can do it. And your child will do a lot of self-direction once he gets started.
Finally, there can be math competents. My daughters are math competents. Math competents do not especially love math, and they can lead a happy life without it. However, they have been taught math correctly. They are good at math (especially compared to incompetents), and they enjoy it.
Home Schooling, Tutoring, and Teaching a Class
This method is ideal for home schooling and tutoring one-on-one.
For larger groups, and even just more than one student, it is very useful to put the problems you plan to use on paper. Then the students can work on their own, at their own pace. Also, you automatically have a record of their progress. For a few students, each problem can be on its own sheet of paper. For larger groups, you will have to put sets of problems on a single page.
One more small point -- if you are math incompetent, you will tend to think that the most interesting math problems are the ones that in fact are the least mathematical. This website contains problems designed not only to teach math, but to be interesting to math lovers.
Content is important, but..... A wise man once told me there were three things people had to learn -- social skills, communication skills, and problem-solving skills. Math is an excellent medium for teaching problem-solving skills. I feel that should be the main goal in teaching math.
However, you will do better if your main goal is having your students enjoy themselves. This really is the best goal, and it also happens to correspond to learning. If your student is having a bad day, give up. Your student isn't going to learn anything, and a lost day is nothing. If your child hates math, give up. Your child won't learn math. The goal is to teach problem solving skills -- find problems your child enjoys.
Being the Guide
You can monitor the emotional/cognitive aspect of this learning. Sometimes you will give a problem that is too hard. Fine, no problem. If you were a wiz, you could maybe construct problems ad hoc to bridge the gap to the solution of the problem. If you are not, just set the problem aside for another day. Or another year, there is no hurry. If your child does not get the problem right, you can tell your child the answer is wrong and to try again. But you usually don't have to tell your child about the error.
If the problem is too easy, that usually is not a difficulty either. A constant barrage of difficult challenging problems is usually not right, there has to be some easy problems. If your child is bored, then the problems are too easy. You are in an ideal situation to monitor that. Meanwhile, your child will also probably do a brilliant job of selecting problems to learn from. In reality, a child will find some problems interesting and some problems uninteresting. That will correspond reasonably closely to which problems your child is learning from.
These problems should be fun. To learn, your child must do some hard thinking. But finding the solution should be fun. If your child is not having any fun, this method is not working. There are more important things than math. Maybe your child is worried about something else. Maybe your child just doesn't get it. You can lead a horse to food, but you can't jam it down its throat if it doesn't want to eat.