Some things are presented to students as if they needed to be taught, learned, and tested, when the teacher could instead find strategies that allow children to experience them, become familiar with them, and ultimately take them for granted. That alternative could be called teaching number sense. When that has happened, a time may come when children can be helped to formulate what they already know and practice in words that could be seen as rules. A determination to expect the knowledge to be imparted by the rules, as opposed to trusting that it can be experienced and taken for granted, can be counter-productive. All teachers practice the two approaches to some degree, but the emphasis is seldom as heavily weighted as it could be on handing knowledge over to a child’s intuitive understanding. Let me briefly look at Place Value as an example of this perspective.
Place Value represents an essential number sense understanding, indispensable just about any time children and adults manipulate numbers. It is also one of the topics that can take hold quite naturally instead of being allowed to build-up into some major hurdle, from the early implementation with addition and subtraction, to its application to decimals, multiplication of decimals, and connections to fractions and percentages.
I just said that place value was something that could take hold quite naturally. That is said from the perspective of the child, not the teacher. From the teacher’s perspective, fostering in the child that natural understanding may take careful and detailed planning and a determination not to miss any opportunity, however small. It may imply devising a number of strategies that are not just implemented on a few occasions when that is the topic of the hour, but on a constant basis as it is a constant concern in the mind of the teacher, including extensive practice adding and subtracting dollars and cents. Let me expand on two of them.
Of course, we want students to align decimal points when adding. We also want them to see the symmetry as we move left or right that allows them to read 2100 as 2 thousand 1 hundred and 0.012 as representing 1 hundredth and 2 thousandths. But the decimal point is not the axis of symmetry of a number with decimals: to the left of the decimal point we have the ones, and then the tens; to the right we have the tenths and then the hundredths. That is at the heart of a child’s confusion when interpreting hundredths, for instance. In terms of place value, the ones’ digit is the axis of symmetry of a number with decimals. Children can be helped to acquire a vision of the correct axis of symmetry by having them color the ones’ column or circle the ones’ digit of a single number as they learn to interpret the place value of decimals. Now, tens and tenths, hundreds and hundredths are equally distant from the perceived axis. We may also devise simple graphic ways of illustrating the symmetrical connection.
An axis of symmetry belongs equally to both sides. With the unit digit perceived as the axis, the next step is to encourage students to begin with the unit digit as they seek to correctly identify the decimals, in preparation for a world of science that uses them constantly.
- 5,634.92? The mind focuses on “4 hundred and 92 hundredths”, which of course, interpreted as the improper fraction that it is, is 4 units and 92 hundredths, with the two decimals now easily identified as hundredths.
- 267.532? Focusing on the decimals but including the unit digit, we think: “7 thousand, 5 hundred and 32 thousandths.” The mind just needs to drop the 7 of the unit place value to correctly interpret the decimals without any need to count the tenths, hundredths, and thousandths.
A second concern applies to a child’s very first experience with place value. When standards limit the range of numbers a first grade student should practice adding and subtracting to 100 or 120, I immediately conclude that the limit does not represent a concern for teaching place value. Children like large numbers. As soon as they learn to add ‘5 books plus 3 books’, they know how to add ‘5 hundred plus 3 hundred’ or ‘5 thousand plus 3 thousand’. They take great pride in adding such numbers, and one student is very likely to suggest expanding the understanding to millions. Limiting the range of numbers to 120 is essentially asking students to learn first the most difficult numbers, the ones where an understanding of place value is not reflected in the all important oral version of the numbers.
There is a reason why the verb “to be”, the most frequently used word in the English language, is also the most irregular: I am, he is, we are, she was, they were, been, being. It’s a marvel that we recognize all these entirely different words as different versions of the same verb. Such irregularity would be unsustainable with a verb that was used only on occasion. Who would bother to remember all that? The same is true with numbers where the smaller numbers, the ones most frequently used, are also those that can afford to be the most irregular. In their oral version and its spelling, the sequence of numbers up to eleven is purely arbitrary: we cannot imagine the next one from knowledge of the previous ones. ‘Twelve’ has a very thin connection to the ‘two’ of 10 + 2 just as ‘twenty, thirty’ have a very discreet connection with ‘two’ and ‘three’. The logical patterns emerge systematically only with 100 and larger numbers.
By starting with the oral version of large numbers, and gradually moving back to digit numbers and to numbers smaller than 100, we are only following the basic strategy of moving from what is easy to what is less so. We are in the process transferring the ease and understanding, not just the facts themselves. I would want to do this in the early grades without ever referring to ‘place value.’ Putting numbers larger than 100 or 120 off-limit in first grade robs us of one strategy for allowing children to grow into a natural understanding of place value.
On this and other topics we may be satisfied with letting children become very familiar with implementations that are very easy and almost intuitive, and finding strategies to do so. We may then find that acquiring the knowledge of the more difficult applications no longer represents such of a challenge.