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).

General Approach to Learning Math

While the CCM Standards for Mathematical Practice may be useful to assess which students are mathematically proficient, I find them abstract. My approach to learning math includes five general suggestions in this post, and a five-step problem-solving process in the next post.

1. Learn the definitions

To learn math, we must learn definitions. New words just keep on coming in math, from the earliest arithmetic – count, add, subtract – to advanced math – differentiate, prove, transform, etc. Learning the exact meaning of those words is essential for any student of math.

2. Discover and use math in hobbies and jobs

This is related to CCM Standards Step 4, but it is goal-directed. There is math in sports, shopping, cooking, baking, traveling, gardening, etc. These and countless other areas of interest offer opportunities to reinforce math learning. To continually strengthen our skills, it helps to use math daily. I will devote a chapter to ways to do this, but you and your children should find it fun to add math to activities you already enjoy. For all but the most curious students, this is necessary. Children are too bright and too easily diverted to practice anything without strong motivation.

3. Customer satisfaction

This idea may seem a little offbeat, but it is timeless. Your children may be young, but sooner than you might imagine, they will be in business at some level. I’m sure you know that the success of businesses and their employees reflects customer satisfaction. The customer wants the product to operate as advertised and to be durable, efficient, reliable, cost-effective, and easy to maintain. She wants the service to be cheaper, faster, and better. My job is to exceed customer expectations according to the classic business motto, “under promise, over deliver.”

How is all this relevant to your child learning math in school? I recommend that you help your daughter to view her teacher as the customer. We do not have to love our customers in business, but if we fail to make them feel valued, listen carefully to understand their needs and wants, and meet their expectations for our products and services, they will not be our customers for long. While I would not encourage your daughter to become the “teacher’s pet,” she can and should be respectful and appreciative without being ingratiating. And, if she strives to meet or exceed teacher expectations, she will be successful in school.

An anecdote which recently caught the attention of the anti-CC wing of the Parent Party was viewed by many on the internet: “The first question [on a quiz] asks the student to calculate 5 x 3 using repeated addition. The student wrote 5 + 5 + 5 = 15, and was marked wrong, with the teacher writing in the ‘correct’ solution of 3 + 3 + 3 + 3 + 3 = 15.”  What would you do as the parent of the student in question: would you say, “that’s ridiculous, you got the right answer?” Or would you say, “you need to understand what the teacher is asking for and give her that answer?” I say: “customer satisfaction, understand what the teacher wants and give that to her.”

If you say that the teacher did not have to be so rigid and could have given the child full credit, I will not argue. Since that did not happen, it is okay to ask for the teacher’s rationale; that’s how we learn. In addition to the principle of customer satisfaction, there may be a math lesson here: while it’s true that 3 x 5 = 5 x 3, this commutative property of multiplication is not true for all arithmetic operators. For example, 3 / 5 ≠ 5 / 3. The way we usually write arithmetic expressions (prefix notation) associates the operator with the following operand; however, the teacher may simply be reinforcing good habits regarding observing the order of numbers and symbols. Another example is the expression 3 + 5 x 7 may not be evaluated the same as (3 + 5) x 7.

Bottom line: satisfy the customer (teacher). It is part of the learning process to ask questions, like “why do we have to it this way?” and then listening to the answer. Even in the unlikely event that the teacher answers, “just because” or “because I said so,” she is still the customer. In such cases, it falls to you, the parent-coach, to offer your child a more persuasive answer. I suggest that you do this without challenging the teacher’s authority or competence; it is generally risky to confront your customers and it could cast doubt on the message you are trying to send to your child. Rather, when there is confusion, a good question might just prompt the teacher to clarify his idea.

Is this just subservience? Why am I spending so much ink on this? I believe that striving for customer satisfaction is a lifelong value that transcends formal education in general and math in particular. Your son or daughter will always have bosses: at home, their bosses are parents or guardians; in school, they are teachers; in the military, they are commanding officers; in business, they are supervisors and, from CEO to clean-up crew, they are customers. Treating customers with respect generally pays off.

4. First learn the math, then learn to use current tools

This step is similar to Standard Step 5. Current tools include calculators, computer programs, and spreadsheets. The latter support the use and reuse of data and formulas to improve the efficiency and accuracy of repeated calculations and record-keeping of both data inputs and results. Spreadsheets are valuable tools as witnessed by their wide use in business, education, and sports. I believe that it is helpful to introduce spreadsheets to students sooner rather than later. It is important to understand the theory behind spreadsheets; it is equally important to understand when and how to apply this ubiquitous tool. The spreadsheet learning process works in two directions: a student must understand the arithmetic to program the spreadsheet; and a clear understanding of spreadsheet processes reinforces learning arithmetic.

5. Keep up with the class

Here’s a simple fact that most of you understand very well: if you have done the assigned reading in advance, the in-class lesson will be clearer because 1) the vocabulary, the definitions, and the methods and formulas will be familiar, and 2) any question from the advance reading is likely to be answered in class – and if not, the student will normally have the opportunity to formulate and ask a question. As obvious as this might be to you, it will not be obvious, at least at first, to your children. Reading a lesson in advance, even quickly just for familiarization, is an excellent work habit. Students who do this routinely in college are likely to do well.

Moreover, “flipping the classroom,” a popular concept that may have already come to a school near you, takes the view that homework should be what we now consider “read ahead” and that problem-solving should take place in class: under teacher supervision, at school (once upon a time) or on Zoom™ (in a COVID-19 world).

A real-life situation, such as an accident or illness, can disrupt this sensible process. Often, intervention on your child’s behalf will be necessary. This may seem obvious, but what is less obvious is that you may have to make time to help your child catch up with the class once the emergency has passed. Making the effort sooner rather than later will reduce the probability that your child will fall even further behind.

I suggest that you encourage your children to review past homework and tests. I give teachers full credit for the time they put in to read papers. Students can profit from these investments on their behalf to strengthen future work. This practice can dramatically reduce the occurrence of similar issues in future.

Summary process for a new area

Opinion – Processes matter

This opinion applies directly to the two lessons to follow. In this post, I will discuss the process for addressing a new math domain, such as counting, arithmetic, algebra, or geometry. In the next post, I will put forward a process for solving problems. Each process draws from the practice standards reviewed in the previous 8 posts.

Imagine that you and your family will be visiting, or even moving to, a new place, where people speak a different language and engage in different activities. My guess is that before you go, you will discuss with your family what it will be like, why you are going, what each person should bring, and how they might prepare. If there is anything really scary about the place, such as “they do long division there” (it’s okay to scream), that you would reassure your family that you will be close by to help if needed.

One reason to prepare everyone for the journey is that, as you already know, children have no end of “why” questions. The better informed they are in advance, the more open they are likely to be to the experience. In the case of a new math domain, once they know about the benefits – for example, becoming even more competent at math – they will be eager to begin the journey. Of course, to answer those questions in advance, you will need to do some preparation yourself. Never fear, I am here to help.

CCM Lesson of the day: Process for entering a new math domain

Learn a little about the history and depth of the domain

A simple example of this, which will be repeated, occurs in the domain of counting. As anglophones, we tend to count in English. In spite of a number of anomalies, over the years most of us have come to accept our English-language numbers as “normal.” This is not surprising, but a close examination does reveal irregularities.

·         Where do the names of numbers such as ten, eleven, and twelve come from?

·         Why does the spelling change from three and five to thirteen and fifteen?

·         And from two, three, four, five to twenty, thirty, forty, fifty?

·         And then there is zero – is that even a number?

More. When learning to read, children are confronted with two, which sounds a lot like too and to. In addition, as you well know, children may also hear people counting in other languages, even in your own family. There are other examples, some of which will be highlighted when we get to the counting domain.

Learn the definitions

This principle repeats and repeats and repeats. I found it a little shocking how much of math, especially advanced math, comes down to definitions. I consider definitions to be shortcuts for describing complex math structures. For example, once we agree that plus means to add two quantities, I can say five plus seven and you know what I mean.

Teach to learn

Teaching helps two people: to teach a subject, a teacher must learn it well; then, the person being taught learns from the teacher. In the case where a child is teaching a sibling – usually but not always younger – the coach (you) can make sure that each child benefits, an excellent win-win.

Use math in routine activities to reinforce concepts and operations.

This principle offers repeated evidence of the utility of the math that applies to the activities as well as practice in using the math. Using math regularly fosters respect and appreciation for math based on objective results rather than opinion.

Learn to use math tools, particularly calculators and spreadsheets.

Tools help us in several ways:

·         they remind us that our primary job as mathematicians is to formulate problems correctly.

·         they can save time by performing repetitious operations rapidly.

·         given a correct formulation of the problem, they yield correct solutions.

·         they help us to achieve precision; and

·         they allow us to verify results generated in other ways.

Memorization is a part of mathematics

Accurate memorization is a time saver in math. Memorize the numbers, their spelling, and their numerical representation. Memorize single-digit addition and multiplication. Derive and approximate value of Pi (π) and then learn it to 5 decimal places. Derive and memorize the quadratic formula and Pythagoras’ Theorem. Those and many others can be helpful when you are away from a computer or other tools.