Math: Drilling or Killing?
"Could you tell me the name of the math drill book you carry?" asked a woman by email.
"Ugh." I cringed, quite dismayed that someone could even think I would actually carry such an item in my inventory.
As the former owner of a mail-order educational company, I thought I had done my best to get across the point that I don't much believe in math drill for the memorization of the so-called "math facts."
Of course, people are appalled by statements like that. How can I possibly expect my kids to succeed in life if they don't know their math facts?
Perhaps I have been unclear in all my ranting about drill-to-kill programs and even in founding the fun-filled internet support and mocking group, the Gag-Me-With-a-Saxon Coalition. So let me try to explain…
Who
If I read one more article about or hear one more reference to "subjects that your kids don't like, such as math," I will choke! Why do so many children hate mathematics? Why are so many parents (and, I'm sad to say, especially mothers) terrified and/or annoyed at the prospect of teaching it? Is it because it is inherently uninteresting? Is it because only a select dozen or so folks in the world have "mathematical minds" that can comprehend its depths?
My dad has a PhD in mathematics; his specialty is ring theory. I'm 36 years old and I have no idea what that means…and I'm willing to bet most of you don't either. But to love and understand mathematics does not require any of us (nor our kids) to be in that elite dozen or so in the world that I'm sure my dad is in. Every child (and parent too) can love and feel confident in math! And every child can find the beauty in it as well.
What
When asked if my kids know all their "math facts," I generally respond with the question, "What do you include in the set which you describe as the 'math facts'?" I admit that I do this for my own amusement, but it's a valid question. And the usual reply is a look of stupor. Folks can't believe that I don't know what the math facts are—but they usually can't fill me in on the details either.
After getting over the initial shock, some will state that the math facts would be multiplication tables from 1x1=1 up to 12x12=144. When I ask why they don't include 0x0=0 or 13x13=169, they have no idea. Others will state that the "facts" include all single digit addition, subtraction, and multiplication. Some others go well beyond single digit. Some include division; some don't.
Folks seem not to realize that the set of equations they have identified were chosen arbitrarily by someone on a day when they didn't have enough important stuff to do. So before you dive in with forced memorization, it's best to decide what is really worth memorizing in the first place.
When
When your set of important "math facts" is defined, should you jump in with your stopwatch and start the drill? No! If a child is using math, he or she will come to a point where they can see for themselves how handy it would be to have basic addition, subtraction, multiplication, and division "facts" right at their fingertips, so to speak. You won't have to bribe, threaten, or coerce; the motivation for the practice will be intrinsic.
As my dad pointed out, a piano player practices scales, knowing that it will improve his concerto; a tennis pro concentrates on serving, knowing it will improve her game. In the same way a student of math should study the "facts," already knowing the benefit that will be derived, not as an end in and of itself.
In addition, focusing on the "facts" should not be done at the same time that a child is initially studying an operation. As Marilyn Burns
A premature focus gives weight to rote memorization, rather than keeping the emphasis on developing understanding of a new idea. The danger to avoid is having children believe that giving quick, right answers is really what is most important in learning arithmetic.
Where
A few years ago a woman dropped by to pick up some books she had ordered. Thinking, I suppose, that it would be impressive, she drilled her fifth-grade son in my living room.
"What's 8x6, Herman?"
"42!" he called out. "Um, no, 48…no, 44…" After a few more questions with similar results, the agitated mother gave up.
The poor kid could remember the products he had memorized—he just couldn't remember which equation each one fit into!
Practice math anywhere you'd like, but don't make your kids "perform" their math in public unless, of course, they want to.
Why
Emphasizing comprehension of mathematical operations is of the utmost importance, and often sadly lacking in mathematics programs. Thy "why" should be emphasized, not the "how."
In the Miquon Math series, which we use in 1st through 3rd grades, the student learns concepts concretely before learning the abstraction on paper. Standard algorithms aren't taught as "the right way," to do math. Kids often discover their own algorithms.
Regrouping (or borrowing and carrying, in my day) is introduced in the last book, even though multiple digit addition and subtraction are introduced earlier. My oldest daughter came across a problem that said: 800 - 308 = n. Not knowing the standard "cross out the zero" borrowing algorithm, she solved it like this:
I didn't have any units, so I couldn't take away 8 units. So I took 1 of the hundreds from the 800. Leaving 7 hundreds, 0 tens, and 100 units to subtract the 308 from.
I took 8 units from the 100 units and that left me with 92 units. So I put two of the units in the units column and the put 90 units over in the tens column as 9 tens.
Then I took 0 tens away from the 9 tens and that left me with 9 tens in the tens column.
Then I took 3 hundreds away from the 7 hundreds and that left me with 4 hundreds.
This is not the standard regrouping algorithm. But it is a perfectly correct algorithm! And she could figure it out because she knew what she was doing. She hadn't just memorized a trick that meant nothing to her. She knew the "why" of subtraction and place value.
Far too often our children are exposed to math as if it were a bunch of answers and rules and tricks to memorize and then later stick into the right problem. I discuss this at length in my article titled Learning to Love Math. It's not only frustrating for the kids as they reach memorization overload, it also keeps most of them from really enjoying math and from succeeding at upper-level math courses.
How
As a child advances in mathematics, it will invariably be helpful to have basic arithmetic down cold. Dr. Hal G Moore (a.k.a. Daddy) said:
...algebra manipulations and trig identities are what are hard about Calculus, not the integration and differentiation themselves. We expect everyone to know how to reduce a complex fraction without having to think hard about it.
It is difficult for anyone to move ahead in math with any efficiency unless they have a working knowledge of basic arithmetic.
So how do you practice the "facts" without putting the math-haters curse on your kids? Once the understanding of operations and the motivation to practice are there, the options are almost endless. Just spend some time with your child to find out what they would like.
The very best way to practice is through real-life use! Anything that encourages them to use basic math enough to remember it will do. (Don't all kids know what 1 + 1 is? Sure, because they use it all the time.) Carpentry, cooking, sewing, computing, shopping, music, sports, travel, etc. Most hobbies and vocations have some mathematical aspect that could be emphasized.
If the real world doesn't afford enough opportunities there are many other avenues. Computer programs are great for drill (though not usually for comprehension!), as they are usually entertaining enough to make the repetition nearly painless.
There are a number of board games that use lots of math. We use a homemade game my sister and her sons made up a number of years ago and I highly recommend the Muggins Math games. As their mathematical understanding grows, so do the options they have added to the game.
Some kids don't mind flashcards, but I'd only use them at the child's request. I have a gut feeling that many a math-anxious adult started out with flashcards in his face!
Every once in a while we even use Calculadders, a short, timed arithmetic drill. I'd be cautious about using this for some children but, because each set is so short, it has been relatively painless.
A fairly recent invention in our home is Bunny Math. This silly game came about when one of my children asked if I would help her to quickly recall the single-digit times tables. She grabbed a large, stuffed rabbit. We sat cross-legged on the floor, knee-to-knee. I shot multiplication problems at her as quickly as possible. She had five seconds to produce the correct answer before the bunny bonked her on the head. This game has become far too popular in our home as it is requested long before our children have any idea what numbers are and long after they have any need for arithmetic drill. But it is guaranteed to produce gails of laughter. Even with teens.
The method is not terribly important as long as the child continues to enjoy the learning process.
Trick or Treat
Years ago, when my oldest daughter was seven, my neighbor (who has a daughter the same age) said, "You know, Jessica knows her times tables! I drilled her!!"
Funnier than the fact that my neighbor felt compelled to drill "the homeschooled kid" on math while she was playing dolls, was the fact that she so openly admitted it to me! Still, it was quite a revelation, since at that time Jessica had never been timed or drilled or quizzed on anything mathematical.
When asked what her trick was, Jessica replied, "There are no tricks in math. It's only a trick if you don't understand how it works."
Amen!
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