Archers For The Lord: Ask Mary
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How important is it to memorize the Multiplication Tables?
A reader asked, "How important is it for a fifth grader to have memorized his multiplication tables? I try and try, and one minute it looks like he finally knows them, and then two weeks later he has completely forgotten everything again!"
It is far more important that children understand what they are doing, rather than simply memorizing tables. Young children, in particular, need a lot of hands-on experiences. As many of you know, if you have read my booklet, "Taking the Frustration out of Math", I would personally recommend that you not even USE curriculum materials until about the fourth or fifth grade. Before that time, real life math experiences are best.
For example, rather than doing exercises in a workbook that stress time, money, or measurement, use real-life experiences instead. Time can best be learned gradually by having access to real watches and clocks. Digital is probably best in the early years, since an old-fashioned clock face uses very abstract concepts. "The hand goes from one to two and that means five minutes." ??????? Measurement can be accomplished in the garden or the kitchen or even at the grocery store, where produce can be weighed. Money is best learned through such experiences as having and managing allowances, learning to tithe, having long and short term savings jars, and having a bit of spending money. Old-fashioned "lemonade stands" or other businesses for young entrepreneurs also provide valuable math experiences.
For those who enjoy workbooks at younger ages, I would recommend using something simple, bright, and colorful, such as sticker books from K-Mart or other discount stores. "Manipulatives" can be popsickle sticks, cheerios, or M&Ms. With these, children can learn to count, sort, add, subtract, and learn about the concept of sets. I would also add in some math games in the early years, without too much of an emphasis on "going by the rules"-- allowing a bit of space for creative use of the materials. Some of our favorites have been Muggins/Knock-Out from Old-Fashioned Products; Cathy Duffy's Math Mouse Games (now out of print---watch for used curriculum sales), the projects in the "Family Math" book, and "Baseball Math". The latter also seems to be out of print, but I've seen one called "Slugger" that is somewhat similar.
I also highly recommend the presence of a simple abacus in the home for young children to play with. They can learn concepts such as even and odd, and grouping, through just fooling around with an abacus all by themselves.
When children are ready to learn multiplication, it is important to make sure they understand that 3 x 2 means three groups of two--.Again, use cheerios, m&ms, (or, if you must, expensive plastic things) to actually arrange items into groups. Help them to understand the "commutative property of multiplication" by showing them that three groups of two and two groups of three wind up with the same number of objects. Once they have a solid grasp of this, (and it might take a long time!), start out with skip counting--a great project for the car if you aren't too pushy.
To skip count, take a multiplication table, such as the 6s, and figure out what 6 x 12 is. (72, for the mathematically challenged among you). Then have them count by 6s up to 72--6,12,18,24,30,36,42,48,54,60,66,72--Then backwards by 6s, which will be harder. Do it many times, not just once. Now point out that those numbers are going to be the answers to the 6s multiplication tables. Then start with 6x1 is 6, 6x2 is 12, etc--.Only when they are very adept at this, begin skipping around, asking questions out of order. This entire process, for all the times tables from 1 through 12, can take a couple of years if done right--I'd say this is roughly appropriate for fourth or fifth grade level, but it depends a lot on the individual. Some kids can do it much earlier, some will still be having trouble in sixth grade.
After they thoroughly understand multiplication, it is time to have them make a multiplication chart, which contains all of the multiplication facts through the twelve times table. Make sure they do it themselves, with a few instructions from you. At this point, they will start noticing some of the patterns, which will help them memorize their tables all the way. When they completely understand the tables, the next step is to use "Learning Wrap Ups", which are self-graded, to improve their speed--but only use stop watches if THEY want to use them. There is no point to creating additional stress on kids who have some difficulty with math. I personally don't like flash cards because it introduces your "relationship" into the equation, i.e. it isn't good when you want to kill them halfway through the session.
The important thing is not EXACTLY when all this is accomplished--but that it is done in the right order, emphasizing understanding, not rote memorization. Also, don't allow them to use calculators until calculators are totally unnecessary--somewhere around the beginning of algebra. An exception might be to allow them to use the calculators to check their work. I also do not believe in the concept of teaching "tricks" UNTIL you are sure they understand the actual process. (An example would be all the tricks associated with the nines multiplication table.) On the other hand, if they discover the tricks themselves, that is actually helpful to their developing understanding.
It generally isn't necessary or productive to spend hours and hours on math every week, doing a lot of repetitions. A little work done on a regular basis to help with understanding will do a lot more, in the long run, than a lot of busy work designed to reinforce concepts they don't really understand. Also, it is helpful to include story problems, especially asking them to write their own story problems. If they can't do it on their own level, have them write story problems for younger students. If they "can do their math", but "hate story problems", it is a sure sign that they are using tricks and memorization, rather than understanding what they are doing, and probably means it's time to back up and slow down for awhile.
As far as I'm concerned, the only time memory and speed become essential is when the kids get to algebra. At that point, having to stop and figure out 6 x 7 will slow them down to the point they won't be able to do a good job. Finally, I personally don't believe in the concept of "pre-algebra"--Get the kids good and solid on all the basics, including decimals, fractions, percents, ratios, and proportions, and wait until they can think abstractly, which varies from about age 13 to 15, and then go directly to algebra. If your kids have a great deal of difficulty, take heart! I was the world's most pathetic math person in high school, and wound up teaching pre-calculus. So don't despair. If I could do it, they will be able to catch up, too, when they finally get ready!
At what age should we begin foreign language instruction?
At what age should we begin foreign language instruction? Should we insist that our children learn Spanish, since it is used by so many people in our area down south, or should we let them learn Japanese? My ten year old is interested in learning Japanese because of his interest in video games.
There are a number of factors to consider when considering studying a foreign language. I was raised in a German environment. My grandfather lived at a German old-age home, and my mother frequently spoke German around the house. I never really "studied" the language, except for a one-year stint in high school, when I had little motivation for learning any academic subjects. To this day, although I really am not capable of reading, speaking, or writing German, I can understand most of what people are saying when I'm confronted with German speakers, either in person or on the television. Clearly, early learning is very potent. The little bit of German I do know is very second-nature to me and doesn't even feel like a foreign language. However, I believe that the fact it was a natural part of my environment, rather than a "learning experience" made the learning stick more than it would have if it had been part of a "unit study in German".
When my own children were young, I introduced them to the sounds of many languages. The only time we ever looked a bit like a school was at the breakfast table. I figured I had them all there anyway, so we might as well do something! We read "The Story Bible" by Pearl Buck over and over for years and the kids did their own artwork to illustrate the stories from the Old and New Testaments. Afterwards, we always did one other "subject". A good deal of the time, that subject was a foreign language. We studied French, Spanish, German, and a little bit of Russian. (Actually, I was the one who wanted to learn the Russian, and I backed off after it became obvious that I was the ONLY one who wanted to learn the Russian!)
One of the reasons I did this was to train their ears to hear the sounds of other languages. Obviously, to accomplish that, it was important that I was pretty good at pronunciation myself, or I would have had to use other resources. At that point, we certainly weren't striving for fluency. I also tried to tie in information about other cultures, religions, and peoples. I believe that these early experiences laid the foundation for later work in foreign languages.
When learning a foreign language, self-motivation is extremely important. Right now, I'm trying to increase my own fluency in Spanish and learn Korean on an intermediate level. My motivation for learning Spanish is to communicate with the many Hispanics in our area, and to be able to help Hispanic mothers consider the possibility of homeschooling their children. I'm learning Korean because two of my own kids live in Korea, I'm visiting there this fall and doing a workshop in Seoul, and two of my books, "The Relaxed Home School", and "The Joyful Home Schooler" are being translated and published in Korea sometime soon. Those are powerful motivators. Back when I was trying to learn Russian, the only real "reason" I had is that I thought it sounded cool. That wasn't nearly enough motivation to persevere when things got tough, so I wound up giving up after awhile.
When trying to learn a foreign language, it also helps if you have people or situations in which to practice. In many areas of the United States, Spanish is best for this purpose, because you have your choice of watching Spanish television, listening to Spanish radio shows, or going out and finding real Hispanics with which to converse. It is usually tougher finding similar opportunities to practice other languages.
That being said, however, I believe in allowing children to study any foreign language which they would like to learn, since internal motivation is so important. Definitely, in the case of your children, if they are motivated to learn Japanese, let them. However, don't feel like YOU have to learn Japanese if you'd rather learn Spanish! The challenge, of course, will be to find the people, experiences, etc., which will help them to learn, especially if you don't want to put in the time to learn the same language yourself.
I have personally made the most headway through using audio programs in my car. Getting into some kind of a habit where the study of a foreign language is a normal part of your daily routine is critical. I've set tons of goals in the past few years that never happened. My goals for foreign language study have been met because EVERY time I go out in the car, I turn on a tape. (The series I like is Pimsleur audio, available through The Ultimate Language Store in Dallas TX,cheappimsleur.com). (P. S. Not really cheap.) I also try hard to either watch a few minutes of Spanish television at night (I prefer the news or weather, because I recognize more words in those situations), and I spend a little bit of time reading in Spanish each day, especially the local Hispanic newspaper. Another thing I've done is to get several books in both Spanish and English, such as Joel Osteen's "Your Best Life Today/Su Mejor Vida Ahora", and read them one paragraph at a time, first in Spanish and then in English.
I believe that learning at least one foreign language makes a person more well-rounded. The only better thing is to travel abroad. I know my recent trip to Africa was a life-changing event, and I hope to continue growing as I begin doing more speaking outside the U.S., now that my own children are grown.
Finally, as a "relaxed homeschooler", I never really worked with my kids on English grammar when they were young, preferring to emphasize reading and writing in a natural manner. When they began studying foreign languages, often at the middle school level, it provided a perfect opportunity to learn some things about English grammar that they had never learned before. For example, when they started learning how to conjugate verbs, they realized that English has a "hole" in the language, because we have no second person plural pronoun. That's why each area makes up its own y'all, youse guys, yo'uns, etc. To me, that was a more interesting way to learn grammar, when comparing one language with another.
If you have one student who has no internal motivation to learn a language, and you are simply doing it to get it on a high school transcript, I recommend doing it the last two years of high school, because they may have to pass a placement test, and if they have zero motivation, they won't remember it for long. However, if they are motivated at any age, the learning should be much better and longer lasting.