> On Wed, Dec 23, 2009 at 11:10 AM, Bill Marsh > <email@example.com> wrote:
> Real Numbers in the Primary Grades > Bill Marsh > Winter Solstice, 2009
This was an interesting post Bill. Thanks for sharing it; I enjoyed studying it.
> Professor Wu introduces real numbers geometrically, > the way the Greeks did, after they discovered that > fractions weren?t up to the job of naming all the > points on lines. He has answered questions asked a > century ago by Henri Poincare in Science and > Hypothesis: "[D]oes not the difficulty begin with > fractions? Should we have ? a notion of these numbers > if we did not previously know a matter which we > conceive of as infinitely divisible ? i.e., as a > continuum??
As noted recently, I was introduced to Caleb Gattegno's work by one Dr. Benson (Stanford).
Gattegno was not just some small time player. He started the UK's version of the NCTM.
> Real numbers are easier to understand than > fractions. Children arrive at school interested in > measuring and can understand and use "measuring > numbers" in second grade as "good names for dots on > number lines." Adults can understand real numbers as > infinite decimals that specify a point on the number > line.
> Remembering that computers use binary arithmetic, > with just the "bits" zero and one, you can learn how > computers represent real numbers right now.
> To interpret something like .1010? in binary on a > number line (which we think of as going from left to > right), start with your hands, palms inward, about two > feet apart and imagine that your left hand is at zero > on the number line and your right hand at one. Every > time you hear or see a zero, move your right hand to > what was the midpoint between your two hands. When you > see or hear a one, move your left hand to the > midpoint. You can increase the sense of zeroing in on > a point, if you want to, by imagining moving forward a > little each time you move one of your hands.
> Since I don?t want to confuse children with two > meanings for .1010, I have second graders use L instead > of zero and M instead of one, saying them as "less" and > "more" respectively. They understand L as meaning less > than halfway and M more than halfway in whatever > interval you are in. They later learn that the > intervals include their endpoints.
This will remind some students of the game of "hotter" "colder", if they've ever learned to play it. Learning such games at a young age *does* have its applications. Gradients in a vector field....
Even if real numbers are "easier" to understand, we don't have the luxury of dropping rationals, so it's a matter of where to start and how to proceed.
On our digital math track (so-called), we build an awareness of "types", which means what it sounds like: types of object or thing.
Most computer languages have types, and it helps organize thinking to look at a Float type versus an Integer type versus a complex number type versus an alphanumeric String or whatever.
Vectors and Polynomials are also "types of math object" which we're free to express concretely using some "dot notation". Dot notation expresses noun.verb(arg) and noun.attribute grammar.
A weakness of most math curricula is they neglect to share (a) overview (big picture) and (b) lore.
Overview is like preview, supplies a road map.
For example I see no reason to hide this little mnemonic from first year algebra students: N < Z < Q < R < C. One might fancy it up a bit, use a more formal subset symbol, split out the wholes from the naturals (counting numbers), but that's an important Venn Diagram (concentric circles OK).
Lore means adding an historical dimension and actually supplying some context and motivation by this means.
Lore = Stories. Like the story of zero.
I'm not against having purposely inane, demented and/or fictional story problems, but not exclusively.
"How things work" is at the very least an important kind of story. Just as important are stories about civilizations and their ability to adapt, innovate (or not).
Kids play these simulations for fun anyway. Lets make it more real, connect more dots. Yes, values enter into it.
> Third graders and the rest of us can understand and > use decimals in this same zeroing in way. We can use > the notation 3.14_ to denote the closed interval that > we usually denote by [3.14, 3.15]. This interval is > both the set of real numbers between and including the > two endpoints and the set of all numbers with infinite > decimals that begin with 3.14. Thus it turns out that > the point we denote with 3.14 is the left end of the > interval 3.14_, the smallest number in that interval. > This fits with the way we often say "I?m twenty-one" to > mean I am at least twenty-one.
> Children can primed in preschool for saying and then > understanding decimals a digit at a time. For example, > they can read 3.141592 aloud a digit at a time from > left to right, as good practice in recognizing and > saying digits. People who prefer saying and thinking > of decimals as fractions can do so with this example.
We should be honest in our digital math, and admit that we don't really use a real number type. No computer has ever used a non-terminating irrational in any number crunching by definition. Memory is finite.
"The ontological status of infinity" is maybe too deep a topic for grader schoolers, but their teachers should feel free to immerse themselves in the relevant philosophical debates, at least at some point in their training. Wittgenstein anyone?
> Precalculus students in high school can follow up on > this way of doing things by defining real numbers via > nested intervals, rather than using Dedekind cuts or > Cauchy sequences, as is more common. This was how I > first saw real numbers treated in a completely formal > and logically rigorous way, in high school in 1957. I > can still remember my excitement when I read the > following on page 5 of William K. Morrill?s Calculus.
> "Since [the square root of two] is not a rational > number, how do we describe it? First we pick the > greatest counting number whose square is less than or > equal to 2. Then the number we are attempting to > define must lie within the interval whose end points > are the counting numbers 1 and 2. ...The length of > this interval is one. Now consider the greatest > rational number to tenths whose square is less than or > equal to 2 and the smallest whose square is greater. > ... Continuing indefinitely, we obtain a collection of > intervals with rational end points each of which > contains its successor and whose lengths approach > zero. ... > "The nested sequence of intervals [1, 2], [1.4, > 1.5], [1.41, 1.42], [1.414, 1.415], ? closes down on a > unique number which we define to be [the square root of > 2]. We shall call this number a real number. We have > already seen that it is not a rational number, and so, > to classify it better, we shall call [it] a real > irrational number."
I like the "progressive approximation" approach, with error term implied (a plus/minus or degree of uncertainty). That's characteristic of what goes on in the field, whether or not we have an infinitely divisible continuum (the above makes sense in a discrete setting as well).
Back to fractions: what seems alien about some of this digital math stuff is how we'll use three dimensional, mostly non-rectilinear shapes to promote an understanding of fractions.
Dr. Arthur Loeb set the stage half a century ago in a 1965 issue of Math Teacher in an article entitled Remarks on Some Elementary Volume Relationships Between Familiar Solids (Vol 58:15).
Real numbers may be associated with edge lengths, while rational relationships associate with volume. Both types of number (rational and irrational) co-occur in the same geometric vista.
> My interest in using number lines in the early grades > started twenty years ago, when I saw that children can > learn a real number system before they know or care > what 4/9 is or how we say it. I was greatly encouraged > in 1997 when I read in Stanislas Dehaene?s The Number > Sense that children and other animals have and use a > mental number line in their understanding of numbers. > Having eminent mathematicians now support this way of > doing things is gratifying.
There's the formal number line of historically recent invention, and then there's the age-old association of number with length, which is indeed primal and should be built upon.
Rational numbers may be cast as real, so starting with reals in some of our curricula shouldn't be a problem.
> Professor Wu has in effect suggested a Copernican > Revolution in math education: that we put the real > numbers nearer the center in K-4 math and leave > fractions at the periphery for a while. I have > outlined a way to accomplish this.
The gist of my interleaved remarks is
(a) now is a time to experiment, not all jump on the same bandwagon
(b) try studying reals and rationals (fractions) within a shared polyhedral vista why not?
(c) our current standard fare is still too right-brain unfriendly. The number line is good because it's visual. Using shapes make sense too, where we have edges, areas and volumes to work with. Both rationals and irrationals enter in, along with angles, trigonometry.
(d) let's share more lore, tell more true stories, and yes, it's OK for stories to have a moral message
What concerns me about *any* national standards movement, with stimulus money to back it up, is that we're in a difficult period when experimentation is precisely what's needed.
Clamping down, herding all students through a common gateway, is in my mind the antithesis of what's needed. Mono-culture is a recipe for disaster in so many ecosystems. Why not learn from that lesson?
You want those seeds of change when you need them.
NSF supported a Rational Number Project for twenty years. Maybe now is the time for a two year Real Number Project.