I Don't Know Where to Start
22 Jul 2015Problems are essentially of two types: the ones that the student solves relatively quickly, in which case the student complains of boredom, and the ones that the student feels intimidated by, in which case the student says “I don’t know where to start,” gets frustrated, and asks for help. Is there ever some middle ground between boredom and frustration? Some parents are on the quest for the ideal resource, the one that will be so adaptive, that the student will always operate in this middle ground.
The middle ground is the fata morgana of problem solving. In fact, the student will learn only from the problems that are, at some given time, difficult to solve. It is not a tedious process if we know how to handle it! Here is my preferred approach:

the student should not be “shown how it is done”  either as a video presentation or as an explanation from an instructor or parent. This is generally not helpful because it does not provide a separation between the skills that the student already has (which he would do well to practice) and the skills he does not have (which he should learn from this specific problem);

“help” should be given in a specific manner as explained below.
How we help, (i.e. coach) the student:
We read the problem and identify the words and sentence fragments that the student does understand. Every problem must be started from something we do understand about the statement. It is virtually impossible for the student to not understand any of the words used. Concomitantly, we identify the words and sentence fragments that the student does not understand, and explain them. At the end of this process the entire statement should be clear to the student and some previously studied proofs would have been reviewed and refreshed. Only after this, it becomes possible  and, indeed, not that difficult  to solve the problem.
For example, in the statement “how many fourdigit numbers have all even digits and are divisible by 24” we ask the following questions:

What is a fourdigit number? Expected answer: a sequence of four digits in base 10 of which the first is nonzero. If this answer is incomplete, clarify with the student the details that may be missing.

What is an even digit? Expected answer: 0, 2, 4, 6, and 8. Students often miss the 0. If so, explain to them why 0 is even.

Give an example of a fourdigit number with all even digits. Possible answers: 8624, 2086, etc.

Do you know a rule of divisibility by 24? Expected answer: no.

How can we tell if a number is divisible by 24? Expected answer: we factor 24 into primes, we find that our number has to have 3 factors of 2 and one factor of 3. Therefore, we need to make it divisible by 8 and 3, both of them are divisors for which we do have rules of divisibility.

What is the rule of divisibility by 8? Expected answer: the number formed by the last three digits of the given number has to be a multiple of 8. Deviations from this answer have to be corrected, as students often use imprecise language that conveys meanings that are different from the one intended. Prove and review the rule of divisibility by 8.

Do we need to use the rule of divisibility by 8, considering that the number is even already? Expected answer: no, we only need the half of the number to be divisible by 4.

What is the rule of divisibility by 3? Expected answer: the sum of the digits of the number has to be a multiple of 3. If you sense that the student is simply reciting the rule without understanding it, prove the rule again. Ask the student to redo the proof using a different example than the one you used.

Try to provide a single example of a number that satisfies the conditions in the problem. How did you do it? How could you provide another example? Can you think of a possible recipe for generating such numbers? Even a primitive, clumsy recipe can bring us a step closer to the solution!

What do you think is a possible strategy to find the number described by the problem? Expected answer: we should separate the information about the divisibility by 8 from the information about divisibility by 3. For example, first find out how to generate a number that has three even digits and is divisible by 8. Then, pad such a number with another even digit so the sum of the digits is a multiple of 3.

Are there any shortcuts we can take?
During this discussion, we want the student to write down the work. We make suggestions about how the writing could be improved: start at the top of the page, write cleaner, put the numbers in increasing order so they can be counted easier, think before writing, write smaller/larger, etc. We are picky about the form and ask the student to redo the work in a clearer manner if needed. Do not erase  just cross out mistakes!
This is how we teach problem solving, not by simply putting a worksheet in front of a student and telling them to do it! The “happy medium” cannot be found in that way! Even if the magically adaptive worksheet existed  and I have no doubt that it can be designed  it would not help. The goal should not be to adapt the worksheet to the student, but rather to help the student adapt to the question asked.
I think we have a clear answer now to our initial question “how do I start work on this problem?” The student has to start by sorting out which parts of the statement he does understand. It is impossible for the student to not understand any word of it. Slowly, by researching and learning the meaning of the other words, the question asked starts to become clearer and, eventually, steps toward a solution will emerge from this information.
The process of coaching I explained above may seem like protracted and difficult. And it is at first. But, once proper study habits are internalized, it becomes much easier. By using such a method, we are teaching the student how to teach himself, how to pull apart the information into facts he does know and facts he does not know, how to research the facts he does not know, understand them deeply and use them. The coach will not need to go through this process for months and years, throughout the student’s entire evolution. The student will learn how to learn and will become much more able to apply these techniques independently. We, as coaches, have stopped giving him a fish every day and have taught him how to fish for himself.