Ask Mary: Ask Mary

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!