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Decimals

Two basic ideas provide a solid foundation for using decimal notation competently:

  • Decimals provide a systematic way to name "new" numbers between "known" numbers, always in the same way, and to as great precision as one needs. The numbers, themselves, are not new, but to children who know only whole numbers (and a few fractions), these previously unexplored places on the number line are genuinely "new numbers." An activity from Goldenberg (1991)[1], described below in Learning the code: numbers between numbers, gives students a need for numbers that they have not already encountered, and so motivates the invention of decimal notation.
  • Being thoroughly comfortable with the effects of multiplying and dividing by 10, just with whole numbers, lets the student extend the reasoning to numbers that include decimal parts. By choosing to subdivide the interval between two "known" numbers into tenths and name each of the newly marked positions on the number line ("new numbers"), in order, with the digits 1 through 9, the resulting numbers are visually similar to the familiar whole numbers in a way that makes calculation easy.

Please refer to the PowerPoint presentation
Think Math! introduces decimals in a way that is quite different from other programs, and much easier for students to grasp and master quickly. This PowerPoint presents the essential mathematical ideas, the Think Math! approach, and the reasons Think Math! takes this approach. Explanatory notes accompany the slides.

Establishing Order

Learning the code: numbers between numbers

The image of taking a magnifying glass to the number line to "see" finer and finer detail -- numbers between the numbers we already know -- is especially comfortable for computer-facile students who know about "zooming in" to see greater detail.

Seen from a distance, one part of the number line might look like this: Image:NumberlineZoom0.png.

We're more used to having a closer look Image:NumberlineMag200s.png.

The result might look like this:

Image:NumberlineZoom1b.png

The usual way we present the number line involves an even closer look: Image:NumberlineMag250s.png.

Here's what the magnifying glass shows us:

Image:NumberlineZoom2.png

Of course, there's nothing to stop us from using the magnifying glass again to zoom in even further and see more numbers. Image:NumberlineMag251.png

Prior to learning about decimals, students have a complicated system for placing numbers between numbers. We could think of the fractions system as arising from a collection of various-strength magnifying glasses that zoom in on the number line to varying degrees of resolution. A weak enough magnifying glass will double the resolution we get so that we can see halves between the whole numbers: Image:NumberlineMag251h.png A slightly stronger glass would triple the resolution so that we can see three subdivisions between each whole number, showing thirds: Image:NumberlineMag251t.png We could find a magnifying glass that zooms in enough to expand the numberline seventeen-fold, showing seventeen subdivisions between each whole number: seventeenths. But...

By using a glass that magnifies our view of the line by a factor of ten -- exactly the same strength that magnifies Image:NumberlineMag200s.png to show Image:NumberlineZoom0a.png -- we can use the same system for naming the new numbers each time we zoom in. We will be able to subdivide the new magnified space into ten intervals, and use the digits 1 through 9 to name the points that separate them.

Image:NumberlineZoom3.png

And we can zoom in still further, and further, and further, as close as we need for the precision that we want.

AnchorAn activity to introduce this idea

  •  ?? Image:Decimal6_7zoom.png
  •  ?? Image:Decimal6.5_6.6zoom.png
  •  ?? Image:Decimal6.55_6.56zoom.png

Comparing numbers by "spelling": alphabetical order

AnchorAdding and subtracting decimals

Image:numberlinehoteldecimals.png

AnchorComplements and subtraction

AnchorAddition

Multiplying and dividing by 10

AnchorMultiplying decimals

Any-order-any-grouping multiplication and division by 10

Why does the count-the-decimal-laces rule work?

Ignoring the decimal point and making sense of the result

Extending long division to decimals

AnchorMore any-order-any-grouping multiplication and division by 10

AnchorMore ignoring the decimal point and making sense of the result

Why decimal expansions of rational numbers repeat (or terminate)

Language of decimals

What's in a word?

Most Indo-European languages lose a part of the root that appears in full in Latin decem, 'ten'. The Romance languages lose the final m to become French dix (which gives rise to English Dixie), Spanish diez, and so on. The Germanic languages lose decem's middle c and generally change the m to n (or lose it altogether) to become English ten, German zehn (pronounced "tseyn"), Danish ti, and so on. The -teen in words like seventeen and the -ty in words like seventy derive from those Germanic forms.

Closely related words include decimal; December, which used to be the tenth month, following September (7), October (8), November (9), before July and August were introduced to the calendar; decade; decagon; deciliter; decimate; and tithe (Danish ti meaning 'ten' plus the ending -th which creates "tenth" in both English and Danish).

Deca- refers to 10, while deci-- refers to 1/10. Thus deciliter is a tenth of a liter, and decimals (referring to the system for writing numbers between 0 and 1) refers to a system built on tenths, whereas decagon is a polygon with ten sides.

A controversy about the pronunciation of decimal numbers

In the United States, a tradition has arisen in schools that advocates -- and in some districts, even requires -- numbers like 3.12 to be pronounced "three and twelve hundredths" rather than "three point one two" when they are read out loud.

This tradition about pronunciation does not exist outside of schools and, for the most part, it does not exist in other English-speaking countries. Doctors and nurses describe normal body temperature as "ninety eight point six degrees," not "ninety eight and six tenths degrees." Mathematicians pronounce the five-decimal-digit approximation of pi as "three point one four one five nine," not "three and fourteen thousand one-hundred fifty nine hundred thousandths." And engineers, scientists, economists, machinists also use the "a point b c d" pronunciation. So do schools in Australia.Anchor

The rationale for a special pronunciation in schools

  • It emphasizes a legitimate mathematical connection. It is plausible that connecting ideas of decimals with ideas of fractions, from the start, adds meaning or understanding to the new "code" (decimals) that students are learning.[2] The number 3.12 does equal (and is one way of writing) 3 \frac{12}{100}.
  • Avoiding any use of "point" prevents students reading 3.12 as "three point twelve." The "three point twelve" pronunciation is not normative in mathematics (the normative out-of-school pronunciation being "three point one two") and also plausibly contributes to the extremely common misconception that 3.12 follows (is greater than) 3.2 (because "twelve" follows "two"). A non-mathematical use of the notation 3.12 to indicate, for example, the twelfth problem in chapter three -- alas, sometimes in the very same textbooks from which students are learning mathematics! -- does get the pronunciation "three point twelve" and does signify a problem that comes later than problem 3.2. This notation is probably worth avoiding in mathematics texts, at least at and before the levels at which students are learning decimal notation.Anchor

The rationale against a special pronunciation in schools

Deliberately making school "truth" different from real-world "truth" is always risky. In this particular case, there are several specific disadvantages.

  • It blurs the usefulness of decimal notation by treating it as "the same" as fraction notation. There is a good reason why we have two different notations for such familiar quantities as 2/5, 1/2, 2/3, and 3/4. Standard fraction notation makes certain kinds of reasoning and computation easier; decimal notation makes other kinds of reasoning and computation easier.

Notation matters
One of the specific advantages of decimals is that they make magnitude (the order of numbers) easy to determine. It takes computation to decide which of 5/13 and 11/29 is the larger number. But if one knows the decimal "code" (the way those numbers are written), it is tell which of their approximate decimal equivalents, .3846 and .37931, is greater. By contrast, it takes no effort to multiply 1/5 times 1/3 in fraction form, but those same numbers, in decimal form, would be a horrible nuisance to multiply.

  • It teaches a new "language" in a language that children still don't know well. Generally, decimal notation is taught before children are fluent and confident in their understanding and manipulation of fractions. Basing a new idea on an idea that is still imperfectly assimilated is asking for trouble. Decimal notation is a "language" for numbers, as is fraction notation. Teaching decimals to a child by "speaking" fractions is like teaching Spanish to an English-speaking child while speaking German. That's fine, if the child is fluent in German, but a real handicap if the child hardly knows German at all.
  • It exacerbates confusions about place value Though the intent of pronouncing 3.12 as "three and twelve hundredths" is partly to aid in teaching place value notions, that pronunciation may actually add to the confusion. The following two examples, both recorded in classrooms, illustrate the difficulties one can encounter.

The lesson in which this conversation was recorded was about the place value names of the decimal positions -- the notion that in 3.12, the digit 3 is in the "units column," the 1 is in the "tenths column," and the digit 2 is in the "hundredths column." The teacher explicitly reminds the children that they are not to pronounce 3.12 as "three-point-one-two."

Teacher: How do you read this number?
Child: Three and twelve hundredths.
Teacher: Right! How many units does this number represent?
Child: Three?
Teacher: Right! And how many tenths?
Child: (more confidently) One.
Teacher: Yes. And how many hundredths?

At this questions, there is dead silence in the class, no hands raised. The teacher coaxes good-naturedly, something like "Oh, you all know this!" Eventually a child tentatively raises a hand and says:

Child: Twelve???

Well, they've just read it as "three and twelve hundredths." Yet another correct interpretation (but unfortunately not one the children would think of) is 312 (how many hundredths in 3.12 is like how many cents in $3.12; how many x in y is y ÷ x). But the teacher was thinking only of the column, and so was expecting the answer "two."

Teacher: No! There are two hundredths. (She points to the digit 2 and continues.) This is the hundredths place.

In a similar way, forbidding the "spelling pronunciation" sometimes rules out any spoken answer.

The class is doing a lesson on mixed numbers and decimals. Once again students are reminded not to use the spelling pronunciation. The teacher writes 3 5/10 on the board.

Teacher: How do you write this as a decimal?
Kid: Three and five tenths.
Teacher: Yes, but how do you write it?

Finally some kid bravely says "Three, um, decimal point, five?"

Psychology and pedagogy

  • Cognitive development of underlying and related concepts
  • Common difficulties and misconceptions with reference to research
  • Examples of student thinking
  • Visualizations including number line, base ten blocks, pictures
  • Materials and media including teaching and student video, applets, classroom materials
  • Special learning issues including ELL, visual learners, blind, gifted, other specialized populations

History and connections

  • History of decimal notation, international variation in notation
  • Connections to other topics in mathematics, science, social studies, literature (story books), etc.Anchor

References

  1. ↑ Goldenberg, E. Paul. “A Mathematical Conversation With Fourth Graders,” Arithmetic Teacher, 38(8):38-43. 1991.
  2. ↑ Reference needed.