What About Memorizing Math Facts? Aren't They Important?
Well, yes, but the question may be "when," not if, math facts are important.
I'll defer to Frank Smith to help explain.
Julie Brennan, March 2006
A very common anxiety area with elementary math surrounds math fact memorization. Very relaxed methodologies would say, use them and you will eventually learn them. Some more structured individuals feel it is too important a skill to leave to "osmosis" to learn, and begin drill methods early. I tend toward the former, I am willing to relax and let the child learn math facts on their own time table, but I am proactive in communicating the importance that facts be committed to memory for a child to be able to understand and grow in mathematics. I ensure exposure to games and other math fact applications is frequent and full of meaning to the child and prefer these methods to flash cards and other drill methods that lack context.
As my oldest had a particularly difficult time with speedy recall of math facts when he was younger, I've read what I can find on this topic and learned that while conventional wisdom indicates early memorization serves a child well, not all children do manage to remember math facts without there being a higher mathematical context to apply them to. I have four children, and three have had little difficulty with math facts, having learned them through many different activities, but speed of recall was acquired through games. One required more than that, he needed higher level context for facts to stick.
I'd like to quote from Chapter 12, Memorizing, Calculating and Looking Up from Frank Smith's book, The Glass Wall: Why Mathematics Seem Difficult, as he describes what I have observed much better than I can.
"Memory, like understanding, is unavoidable in mathematics and everything else. Although mathematics might seem to be a constant process of 'working things out,' the foundation of any kind of mathematical enterprise is memory, and a great deal of learning mathematics involves committing mathematical facts and procedures to memory. Memory eases all of our way through mathematics, and we can't get started without it. The first conventional step is to memorize, in order, the numbers from one to ten. These have to be known.
The next step in an individual's mathematical development is usually 'tables' for addition and multiplication. I'm not referring to printed tables with column after column of numbers arranged in a systematic way as a substitute for memorization, but packages of knowledge that must be committed to memory . . . often a little bit at a time. Tables in this sense are simply organized ways of remembering mathematical facts." (he goes on to describe various methods young children remember, such as one and one are two, chants, etc.) Once memorized, these facts become immediately available for use provided the nature and manner of use is understood. "A few simple rhymes conceal a detailed blueprint for the foundations of mathematics. . . Many fundamental mathematical and geometrical relationships are woven in the patterns of these tables, when the numbers are read horizontally, vertically, diagonally, or in leaps and bounds. . . "
"How well we remember anything depends on how richly connected it is to other things we know. This is why rote learning is both difficult and inefficient. It is much easier to remember things that are meaningful to us than things that are not. . . A person who understands prime numbers finds it much easier to remember the series 1, 2, 3, 5, 7, 11, 13,... than a person who does not. . . a person who is confused by mathematics usually finds it impossible to remember anything about mathematics. Not only is it easier to put mathematical facts into memory if they make sense to us, but it is also easier to get them out. Meaningfulness, or the richness of connections, determines how quickly and efficiently anything goes into memory, and [out of memory]."
I think this is important to realize when we approach math fact memorization as something to attack separately in a child's mathematical learning. Yes, memorization of facts will be critical to their later success in mathematics. So, how do we provide that context? And how much should be memorized? Finally, when should we begin feeling concern at a child's lack of speed with math facts?
I try to answer the first question in the Nuts and Bolts pages - Games are to math what books are to language according to Charlotte Mason philosophies, and I could not agree more. Games provide real, relevant, connected and fun ways to use math facts repeatedly. When we play a version of War where we add two cards and the winner has the higher sum, you can bet that adding those numbers has relevance :o) Sum Swamp is another early addition game that has been a hit with all my children in learning facts. Classic games like Monopoly have provided many more hours of addition math fact practice.
For the second question, how much should be memorized, I like Frank Smith's answer again. "There's no limit to the capacity of memory, but learning takes time, and recalling can be difficult . . . The limit [on how much to memorize] will differ with every individual. A minimal kit for basic memory would have to include the number 'facts' for addition and multiplication up to 10, plus elementary procedures for simple calculations. . . " That fits with what we all know represents the basic math literacy our children will need to be fully functioning adults in our society. As parents we care, so, the really big question is WHEN to be concerned if our child doesn't seem to be learning their facts. Smith says:
". . . efforts to memorize can be completely counterproductive when we have little understanding of what we are doing. That is one reason why mathematics can suddenly become frustrating and opaque, no matter how highly motivated we are. If we strive to memorize something we don't understand, if we're on the wrong side of the glass wall, we'll have great difficulty trying to remember it. But we'll have no difficulty remembering our failure and frustration - the underlying condition of any phobia."
I think this is the answer to the last question, when. Focusing on developing a child's understanding of number relationships through active, relevant and connected exposure to real mathematics provides this understanding of what we are doing. If they are not commiting facts to memory, investigating where the understanding may be lacking can be a starting point, but my "one out of four children" experience is that for some children, they do need more TIME to develop the kind of mathematical thinking that creates a level of understanding supportive of their recall of math facts in context. Does that mean parents sit passively by? Not in the least. In the case of my one highly analytical and less computationally oriented child, we continued to play games, use math, and not build up anxiety. One day, the speed and recall just stuck.
For some children, gaining this understanding will cause them to commit facts to memory without any conscious drill, if they are permitted the time to learn on their timetable. For others, some additional practice at using facts once this understanding is gained may be desired and is often enjoyed in the context of games or game-like applications.
March 15, 2006