Teaching Math Through Questions and Conversation
Among students around the world, American high school students have one of the highest measures of self–esteem concerning their math abilities. At the same time, the U.S. is one of the lowest–ranking nations in the world in actual math ability. (For a lengthier discussion of this phenomenon, see Mark Bauerlein’s book The Dumbest Generation: How the Digital Age Stupefies Young Americans and Jeopardizes Our Future.) This statistic is both amusing and sobering. As a former engineer, I have a passion for teaching math well. I spend a lot of time tutoring other homeschooled students besides my own sons. Math is one of my favorite subjects to tutor, as it is easy for students to see progress in a short amount of time.
If you ask the average adult if he can count and add, he will say “yes.” To assess his claim, challenge him with a simple problem like this:
1. Can you put these numbers in order from least to greatest?
2. Can you find their sum?
.009 .09011 .000999 .0909 .90909
I worked this kind of problem recently with my 13–year–old son and his cohort in a Classical Conversations math seminar. This kind of problem is always revealing to the students and their parents. About one–third find it easy, one–third need a nudge, and one–third do not even know how to begin the task.
Using the Saxon math curriculum, I have been teaching math to my own children since 1993. As I spent time with them in the elementary years, I over–emphasized their understanding of three critical ideas.
1. What kind of numbers are you using?
2. Which of the four operations are you using?
3. Which of the four laws will you use to discover the answer?
It is critical to establish facility with these questions so that students know how to approach any problem, break it into its component parts, and find a solution. All math courses throughout high school will expect understanding of these concepts.
In order to develop facility with math, students must know how to name the numbers they are working with, such as decimal numbers, fractions, and exponents. They must also be familiar with the four operations: addition, subtraction, multiplication, and division. Finally, they must learn the technical vocabulary or “clue” words associated with each of these four operations: sum, difference, product, and quotient. Then they will know how to start the problem.
Just as students must master the technical vocabulary of math, they also need to learn the laws that govern math. They should begin to memorize the laws before they know how to apply them. My own children and students in my programs practice reciting four mathematical laws every year: the Associative Law, the Commutative Law, the Distributive Law, and the Identity Law.
Now, let us practice by applying these three concepts to the solution of the above problem.
1. What kind of numbers are we using?
Answer: They are all decimal numbers between 0 and 1.
2. Which of the four operations are you using?
Answer: Addition. (Sum is the clue word that indicates addition.)
3. Which laws will you use?
Answer: Associative Law of Addition because I can associate these numbers in any order when I add them.
The Associative Law can be memorized by students as follows: “The Associative Law of Addition states (a + b) + c = a + (b + c).” In other words, I can arrange the numbers in any order without affecting the answer. Young students will recite this law long before they are able to apply it. Then, as they approach problems like our sample problem, they can apply the law and achieve full understanding of it.
In order to add this series of numbers, I must line up the numbers by the decimal and add final zeros to make sure I have done it correctly.
Adding the ending zeros not only helps students to line up the problem correctly, but it also helps to arrange the numbers in order from least to greatest. I tell my students to always think of decimals as money. Because they are experienced with decimals in money, this analogy helps them to see things more clearly. This is how I would phrase it: “Wouldn’t you rather have the most change? The largest number in this sequence (.909090) represents about 90 cents. Or would you rather have the smallest number in this sequence (.009000), which represents about 0 cents?” Point out the first two digits, and things will begin to become clear.
Now, let us return to our original question. Can you put these numbers in order from least to greatest?
Answer: .000999 .009000 .090110 .090900 .909090
These concepts are taught well in Saxon and other popular math texts in the lesson section. Unfortunately, we tend to think the symbolic problems are the most important part of the book and race to complete the problems. Math success results more from thinking about the technical math terms and mathematical laws than it does from writing down numbers. We need to use more precise words and have more conversations about math than we do. One excellent resource for thinking about the fundamental concepts of math is Understanding Mathematics: From Counting to Calculus by Keith I. Kressin.
Keep in mind that Americans—homeschooled students included—have performed poorly in math. This should inspire us to work hard at math every day. As you work on math with your children, use lots of words but also do lots of problems. A focused hour a day on math using the principles above will ensure that your children are prepared for higher levels of math.
Leigh A. Bortins is author of the recently published book The Core: Teaching Your Child the Foundations of Classical Education. In addition, Ms. Bortins is the founder and CEO of Classical Conversations, Inc. and host of the weekly radio show, Leigh! At Lunch. She lectures about the importance of home education nationwide. She lives with her family in West End, North Carolina. To learn more, visit her website, www.classicalconversations.com, or her blog leighbortins.com/blogger
Copyright 2012, used with permission. All rights reserved by author. Originally appeared in the January 2012 issue of The Old Schoolhouse® Magazine, the trade magazine for homeschool families. Read the magazine free at www.TOSMagazine.com or read it on the go and download the free apps at www.TOSApps.com to read the magazine on your mobile devices.