A 5-Step Approach to Problem-Solving

1. Draw pictures (geometric, graphic, maps, etc.), clarify definitions, use abstraction, and estimate answers.

Step 1 of the CCM approach is to “make sense of problems.” Drawing a picture is a good way to begin to solve a problem. Then, we can place symbols (abstractions) on our drawing to allow us to write formulas or equations. Step 1 of my general approach is to “learn the definitions.” This is also essential to problem solving.

Estimation can improve our accuracy and alert us to significant errors. For example, at the end of our restaurant dinner, we estimate that four dinners, two glasses of wine, and two desserts should cost us about $80. If we receive a bill for $220, either there has been a miscalculation, or we were handed the wrong bill. In any event, we will want to review the bidding to find the cause of the discrepancy.

Measurement can be fun. How big is the soccer field at the park? One way to estimate that is to “pace off” the distances. Suppose that your daughter measures the width of her soccer field at 80 paces. How wide is the field? That depends, doesn’t it, on the length of her pace. If a single pace is about 2 feet 6 inches, the answer is… At this point, we recognize the value of terms of approximation such as about, almost, approximately, around, just over, just under, etc.

2. To really learn something, teach someone else

Step 3 of the CCM approach suggests sharing ideas with others. I agree that that learning should be a cooperative – rather than a competitive – endeavor. At first glance this may appear to be altruistic, even self-defeating – why should your son invest valuable time to help another student with whom he may be competing for grades. The answer is that to actually help another student, your child must learn the material well enough to teach it. Showing his fellow student how to solve a problem and then answering questions about it strengthens your child’s understanding. As a tutor, your child learns as much as or more than his fellow student. There is no need or expectation that the favor be returned for the tutor to receive full benefit. By the way, this is what so-called “group math” can be when members of the group teach each other.

Moreover, the competitive aspect of learning is an artifact of school rules, such as, “do your own work.” In “the real world,” knowledge that your child gains is an asset whether or not a colleague also learns it, however the act of teaching increases the probability that such knowledge will be real.

Likewise, at homework time, your child can receive almost identical benefit by teaching you, especially if the definitions, concepts, or problems are new to you. This is a subtle but effective way for you to help your child to exercise a growth mindset to break through a new area, an area that you may or may not be familiar with. One test of this learning process is whether you are able to understand his explanations; for that, you need to be awake and focused. One benefit of this process is that teaching becomes a valuable tool in his learning-to-learn toolbox.

3. Invest time to learn but learn how much time to invest

Fidelity in doing assignments, such as reading the next day’s lesson or solving homework problems, is important. However, I am cautious regarding the emphasis on perseverance in CCM Standard 1. I recommend that you and your child establish a regular time to do homework and a limit on the amount of time to do it. The more time spent on math homework, the less time will be available for other things. Variety is not just “the spice of life,” it is the key to developing multiple capabilities.

It does require some confidence to get past the idea that all homework must be done completely and correctly. Homework is a teaching tool, so using it efficiently is to your child’s advantage for at least three reasons.

There are limits on how much time she can focus on after-school homework; beyond that limit, we are wasting her precious time and energy.

Knowing that the time to finish her homework is limited, her focus and productivity will improve.

And, incomplete homework also has value, especially in math, because it often reveals strengths and weaknesses.

The third point requires her to assume some responsibility. With time running out to complete the homework, she should read any problems not yet attempted. Then, she should write down her questions about the “problem problems.” During the next class, having thought about these problems, she will find the answers to her questions valuable because those hard problems will become solvable problems.

This recommendation is about setting priorities and taking responsibility for those priorities. Not finishing homework is a reflection the amount and level difficulty of assignments and the time available to complete them. Learning to appreciate both the costs and benefits of spending that time wisely is part of learning to learn.

4. Avoid “getting stuck”

Why? Isn’t perseverance an admirable quality, a character trait necessary to success? Not getting stuck is not the same thing as giving up; it simply means that after investing a “reasonable” amount of time to solve a problem and getting nowhere, I may be missing something critical. Ordinarily, if I cannot find my car keys, I have misplaced them. However, on occasion, if I cannot find my car keys it’s because someone borrowed them and failed to put them back. This is something I can only find out by asking. If that is the case, continuing to look in more and more unlikely places will only waste my time.

While perseverance is part of being an effective learner, as was just pointed out (in recommendation 3), effective use of time is also important. As was also just pointed out, homework not only offers an opportunity to practice solving problems, it can reveal what we do not understand. This is valuable information, because it shows where we need to strengthen our knowledge and problem-solving skills.

Here are three things your son can do when he cannot solve a homework problem:

First, ask someone else, a fellow student, a sibling, or you. Even if you are not the one he asks, you can help him learn to ask for help. For example, his older sister may be right person to help him, but some parental oversight may be necessary, at least at first, to assure that help is requested and given in a climate of mutual respect. That may mean suggesting to younger brother that he be ready to explain where he is stuck and what he does not understand. It may also mean suggesting to older sister that her role is to help him to get past the point where he is stuck rather than to solve the problem for him.

Second, some texts offer the possibility of looking up the answer in the back of the book. It takes some ingenuity to use such an answer and then to work backwards. This should not be the usual procedure, because it reduces the value of that answer as a check on the original work; nevertheless, it is a better option than getting stuck.

Third, the process of formulating a good question for the teacher can be helpful since it requires an awareness of where misunderstanding has occurred. At times, composing the question will lead to a redefining the problem, getting “un-stuck,” and finding the solution.

Some parents, especially those whose children are struggling with math, are tempted by after-school tutoring. For example, the question may be “to Kumon™ or not to Kumon?” I say that, unless all else has failed, “no.” The “drill-baby-drill” approach to math is not my idea of how to ignite or maintain enthusiasm for math in your daughter. If you want to extend the limit on the time to do homework, that’s okay as long she – not you – is proposing that. But to subject her to solving hundreds of problems in order to improve test scores is rarely a good investment of time; Kumon is drilling and may or may not promote critical thinking. She has enough math during the school day and in her after-school homework. Better to use the time to work on her two-hand backhand, practice for her guitar lesson, or jog around the lake with her friends.

5. Verify that answers are reasonable and work backwards to verify solutions

Whether on homework, a test, or on the job, we want our calculations to yield correct answers. Unfortunately, since we are all human, we will sometimes make errors. Therefore, we need to check our answers; the more techniques we have to do this, the more likely it is that we will find and correct errors.

For example, having two ways to perform subtraction, the old borrow-when-necessary approach and the new regrouping, our offspring should be able to fix their subtraction errors when they occur. Learning which operations are complementary, such as add/subtract, multiply/divide, and sine/cosine is useful.

Approximation is a tried and true approach to detecting errors in orders of magnitude. Before solving a problem, I can estimate the answer. For example, if 7 students are sick in a class of 150, I estimate that this is between 4% and 5%. When I divide the number of sick students (7) by the class size (150), using a calculator, I get 0.047. According to my estimate, that answer isn’t even close. However, the % sign means percent (per 100), so I need to multiply my result by 100. The calculator was correct, but I failed to enter all the necessary data. Fortunately, my estimate alerted me to my error – the correct answer is 4.7%.

Finally, memorization is generally our friend. Once we understand the theory behind an operation, memorization can be a great convenience. I have seen grocery cashiers who have memorized the code for practically every item in their store; this is a great time saver for them and for their fellow cashiers at customer checkout time when item barcodes are missing. Likewise, your child should know the multiplication tables and use that knowledge whenever convenient. There are many opportunities to memorize useful formulas and operational results (once their meaning is clear).