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.