For many homeschooling families, math is the most intimidating—perhaps even frightening—subject. Many feel ill-equipped to teach high school math courses. Some may even feel nervous about teaching math during the grammar years.
In Classical Conversations, we emphasize skills over subjects. As parents, we need to change our thinking so that we focus on the skills or tools of learning. First, we can approach math as an opportunity to solidify ideas that we once knew but have forgotten. Second, we can approach math as an opportunity to learn new ideas. This is how we redeem the education of two generations at once.
Laying the Proper Foundation in Math
How do we emphasize skills over subjects? We must teach our children how to learn math. In the grammar years (approximately age four through twelve), we need to lay the foundation for upper math study by reinforcing a few key concepts and by learning the vocabulary of math. In the Classical Conversations Foundations program, we memorize the multiplication tables (from 1 x 1 to 15 x 15) followed by geometric formulas and algebraic laws. Students must be fluent in this material before they progress to the more abstract concepts of algebra. One reason American students perform poorly in upper math is because we have forgotten how to lay a firm foundation in the early years.
Two examples should illustrate this point well. When I first began having children, I started a private, in-home tutoring service for high school students. I secretly hoped that I would tutor algebra, one of my favorite courses in high school. Instead, every year the same phenomenon recurred: When the nine-week grades were distributed, I was flooded with calls for geometry tutoring. I could not figure out why so many students struggled in geometry until I began researching the tools of learning which comprise a classical education. I then identified the source of the problem. These students were being asked to perform three very different skills all at once: Memorize formulas (a grammar skill), plug in numbers and solve equations using the new formulas (a dialectic skill), and write geometric proofs demonstrating the validity of the formulas (a rhetoric skill). It is very difficult for the human brain to operate on all of these different levels simultaneously. If the students had memorized these formulas in the grammar years, they would have been prepared to think dialectically with the formulas later in their more advanced studies.
The second example is from home educating my own children. Initially, I questioned the math memory work at the end of each year in Foundations. I wondered why I would ask my six-year-old son to recite: “The distributive law states that a times (opening parenthesis) b plus c (closing parenthesis) equals a times b plus a times c.” [Symbolically, a(b+c) = ab + ac.] First of all, that is a mouthful. Secondly, he did not understand what the words meant. I myself had only a dim memory of algebra class. Then, last year, my son took Saxon Algebra ½. In several lessons, he was asked to identify which laws were represented by given equations. Suddenly, we were both relieved to realize we had already memorized the Associative Law, the Commutative Law, the Distributive Law, and the Identity Law during his years in the Foundations program.
Thus far, we have considered one important idea about math instruction: we must lay the proper foundation for our children. Once that is done, we must master the study skills which allow us to properly learn math alongside our children. In other words, children should first learn the vocabulary of math. Then, we can teach them the study skills needed to tackle upper math.
Learning the Language of Math
Over the years, I have learned how critical it is to teach our children the language of math. Just like Latin, chemistry, or music, math is a foreign language, and we must learn the vocabulary. When you begin working with your children in math, learn the vocabulary and use it often. For example, teach them that two or more numbers which are added together are called addends. The answer to an addition problem is a sum. For subtraction problems, there are three terms: the minuend (the number you are subtracting from), the subtrahend (the number you are subtracting), and the difference (the answer to the problem). In multiplication, the numbers being multiplied are called factors, and the answer to the problem is known as the product. Finally, we come to division. The number being divided is the dividend, the number by which you are dividing is the divisor, and the answer is the quotient.
One way to help your children is to use this vocabulary all the time. When drilling multiplication facts, ask them, “What is the product if the factors are two and six?” If they ask for assistance with a long division problem, ask them to identify the divisor and the dividend before you help them find the quotient. Using the math vocabulary early and often makes it less intimidating.
Finally, we help our students develop good study skills in math by asking questions and training them to either know the answers or to know how to find them. We do not have to know the answers to every math question our students will ask us. Instead, we need to train them to be self-learners through learning the skill of asking appropriate questions. In that sense, our ignorance of the topic at hand can be our best aid because it prevents us from quickly giving them answers when they should be seeking the answers out for themselves.
To illustrate, consider this math conversation.
Child: “Mom, how do you do this problem? Fractions are so confusing.”
Mom: “First, let’s review what we know about fractions. What is the name of the top number?”
Child: “The numerator. I know, the bottom number is called the denominator.”
Mom: “Now, what do you want to do with these fractions, add them or multiply them?”
Child: “Add them.”
Mom: “Yes. Now what do we have to do with the denominator if we want to add fractions?”
Child: “The denominators have to be the same?”
Mom: “Yes. And that’s called . . . ?”
Child: “A common denominator.”
Mom: “Correct. Now, how do we get a common denominator?”
Child: “We multiply the whole fraction by 2/2 or 3/3 or 4/4.”
Mom: “Yes. How do we know what number to use?”
Child: “We find the smallest number that both denominators divide into.”
Mom: “Exactly right. What is the special name of that number?”
Child: “The least common multiple?”
Granted, the child may not get everything right during the conversation. If he cannot remember the special names of these numbers or operations, the student should look them up in the math text. Showing the student how to use the text to look up answers trains him in the important study skill of reviewing previously introduced information.
Mastering the Study Skills of Math
You might be thinking, “This sounds great, but what about the years when the math gets much, much harder?” Hopefully, you have trained your children well in the early years so that they can begin to seek these answers. I have two firm rules in my house which are enforced beginning at age eight. Rule #1: Before children come to ask me a math question, they must attempt to look at previous lessons to find the solution. (Saxon math lessons always list the first lesson in which a concept was introduced right next to each problem. Other math curricula list the important information in boxes at the top of each new lesson.) Rule #2: My students know that they must correct every missed problem in every lesson. There is no point in moving on until we have mastered the material at hand.
Consider this conversation which recently took place at my house:
Child: “I don’t understand this problem. How in the world do I find the volume of this solid cylinder?”
Mom: “First, what is the formula for finding the volume of a cylinder?”
Child: “I don’t know.”
Mom: “Go look it up and come back.” (Notice that I am reinforcing Household Math Rule #1: Look it up.)
[Brief interlude during which child looks up the information.]
Child: “The formula is Bh.”
Mom: “Why is the B capitalized?”
Child: “Because it stands for the area of the base of a 3D solid.”
Mom: “What shape is the base?
Child: “A circle.”
Mom: “Right, so how are you going to calculate the area of the base now that you know it’s a circle?”
Child: “Um . . . the area of a circle would be pi times the radius squared.”
Mom: “Yes. Now calculate that on your paper and label it B. When you have finished that, what else do you need to know?”
Child: “The height of the solid. I multiply that times the base, and then I’m done.”
Mom: “Yes, you’re finished, except you need to include your units. Are they inches, feet, or . . .?”
Child: “Feet. I mean, the units would have to be ‘feet cubed’ once I multiply the area of the base times the height.”
In summary, if we lay the proper foundation with our math memory work, learn to use the vocabulary of math, and train our children in math study skills, we can teach them how to learn math for themselves.