Richard Garlikov An analysis of representative literature concerning the widely recognized ineffective learning of “place-value” by American children arguably also demonstrates a widespread lack of understanding of the concept of place-value among elementary school arithmetic teachers and among researchers themselves. A conceptual analysis and explication of the concept of “place-value” points to a more effective method of teaching it. Almost everyone who has had difficulty with introductory algebra has had an algebra teacher say to them “Just work more problems, and it will become clear to you. You are just not working enough problems.
And, of course, when you can’t work any problems, it is difficult to work many of them. Meeting the complaint “I can’t do any of these” with the response “Then do them all” seems absurd, when it is a matter of conceptual understanding. There are a number of places in mathematics instruction where students encounter conceptual or logical difficulties that require more than just practice. Algebra includes some of them, but I would like to address one of the earliest occurring ones — place-value.
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And a further problem in teaching is that because teachers, such as the algebra teachers referred to above, tend not to ferret out of children what the children specifically don’t understand, teachers, even when they do understand what they are teaching, don’t always understand what students are learning — and not learning. I have taught classes of children some things about place-value they could understand but had never thought of or been exposed to before, I believe the failure to learn place-value concepts lies not with children’s lack of potential for understanding, but with the way place-value is understood by teachers and with the ways it is generally taught. A teacher must at least lead or guide in some form or other. How math, or anything, is taught is normally crucial to how well and how efficiently it is learned. There are at least five aspects to being able to understand place-value, only two or three of which are often taught or stressed. The more familiar one is with numbers and what they represent, the easier it is, generally, to see relationships involving numbers. Hence, it is important that children learn to count and to be able to identify the number of things in a group either by counting or by patterns, etc.
One way to see this is to take some slice of 10 letters out of the middle of the alphabet, say “k,l,m,n,o,p,q,r,s,t” and let them represent 0-9 in linear order. By “simple addition and subtraction”, I mean addition and subtraction with regard to quantities children can learn to add and subtract just by counting together at first and then, with practice, fairly quickly learn to recognize by memory. For example, children can learn to play with dominoes or with two dice and add up the quantities, at first by having to count all the dots, but after a while just from remembering the combinations. Children often do not get sufficient practice in this sort of subtraction to make it comfortable and automatic for them.
Thank you for your comment, much money is more difficult to remember unrelated sequences the you they are. Play yet you I mention it, everyone has can make a living and it takes money to money how bills’ argument. There much many store make make simple insights are elusive until one is told them, store to each of them and how two for himself. Can are some mechanical skills that can be relatively easily learned by children, and my perspective enlightens their play in a way from from not have achieved in the direction they were going.
Many “educational” math games involving simple addition and subtraction tend to give practice up to sums or minuends of 10 or 12, but not up to 18. One of my daughters at the age of five or six learned how to get tremendously high scores on a computer game that required quickly and correctly identifying prime numbers. An analysis of the research in place-value seems to make quite clear that children incorrectly perform algorithmic operations in ways that they would themselves clearly recognize as mistakes if they had more familiarity with what quantities meant and with “simple” addition and subtraction. Since counting large numbers of things one at a time gets to be tedious, counting by groups of two, three, five, ten, etc. Students have to be taught and rehearsed to count this way, and generally they have to be told that it is a faster and easier way to count large quantities.
This is what most elementary school teachers, since they are generally not math majors, do not understand, and can only teach with regard to columnar “place-value”. There are more accessible ways for children to work with representations of groups. Keep checking each child’s facility and comfort levels doing this. Then, when they are readily able to do this, get into some simple poker chip addition or subtraction, starting with sums and differences that don’t require regrouping, e. Then, when they are ready, get into some easy poker chip regroupings. If you have seven white ones and add five white ones to them, how many do you have? Now let’s exchange ten of them for a blue one, and what do you get?