Monday, August 15, 2011

On nothing in particular

Students seem amazed that the idea of "0" as a number is a relatively new concept, but students also see "0" as a numeral, as solid and inevitable as a "3" or a "7"--as most of us do. How could zero be invisible to the Greeks? (It wasn't, of course...but we only think so because we misunderstand our own concept of "0".)

XKCD rules!

That's not how we use it, though, at least not most of the time in the simple measurements or arithmetic we use in class. Zero serves mostly as a placeholder.

Take the number "306"--the "0" means there are no chunks of tens beyond those already captured by the three hundreds signified by the "3". We have 6 units, true, not quite enough to form a chunk of ten, so we need a way to show this: 3_6.

This is glaringly obvious to many of us who grew up without electronic calculators. The calculator we occasionally used in elementary school was the abacus, which requires grasping the concept of placeholders in order to manipulate the beads, and the slide rule, which requires the use of mental placeholders to make any sense at all.

If you do not grasp the very real difference between zero and the other nine digits, subtle problems arise, which become not so subtle when playing with significant figures ("sig figs"), those numerals that carry any real meaning when we apply math to the natural world.

We too often teach sig figs as a set of rules, which make little sense if the student has a poor grasp of zero, which many--through no fault of their own--do. The rules seem arbitrary and arcane, when, in fact, they define the limits of numerical reality when we measure the natural world.

"Numerical reality" comes off a bit too abstract. I should rephrase this: significant figures define the limits of observable mathematical relationships found in the natural world. The relationships are real, inasfar as they can be measured by independent observers. In class, however, sig figs become an exercise in futility for many.

I think it gets down to the zero. Electronic calculators add a level of abstraction to numeracy, leading to the ridiculous assertion science teachers hear at least once a week: "But my calculator says..."

And indeed, the calculator says exactly what it's supposed to say, when its operators plug in numbers without any feeling for what they represent.

And how could they? The Greeks had the concept without the symbol. We, alas, now have the symbol without the concept.

Yes, I know, the Mayans and Indians and Phoenicians and on and on and on had it down pat.
Thank Fibonacci for getting it to the western folk.

XKCD comic used wtihin his guidelines--very broad ones at that. Thanks! 

And yes, the sig fig rules are a tad arbitrary--range of error makes more sense.


David said...

See this is interesting, because I have a strong number sense and I never grew up with an abacus.

I did spend hours and hours playing with numbers with my calculator. I would add/subtract/multiply/divide them and look for patterns. I was able to look for patterns into big, big numbers, because the speed at which I could calculate wasn't limited by the tool I was using.

What's missing from current instruction is actual time using the calculator to play around and construct one's own notion of numbers. Just like you want kids "doing science" they don't get enough time "doing math." The calculations are not the math, anymore than the list of content expected by your state curriculum in science is the science.

There is not enough time spent ensuring kids understand concepts, and too much time kids "cover a broad curriculum."

The issue is not the technology being used to do the calculations, as the abacus is a visual & kinesthetic tool, while the same holds true of the calculator (you do have to press the buttons after all). The issue is that students do not get enough time in math do more than memorize rote facts, and almost never get exploration time.

We both know how key exploration time is in learning.

earlsamuelson said...

I believe the importance of understanding "place value" has been underrated by many. Those who have "number sense" will likely make important connections on their own through exploration. However, for the seemingly increasing number of individuals who struggle with number sense, an effort should be made to help these students make such connections; pattern recognition can once again be employed here. Lattice (grid) multiplication, a method not used often enough perhaps, is very effective in providing a means to visualize place value.

doyle said...

Dear David,

It has long been my belief that those innately curious about numbers, or those who need numbers for whatever purpose, that is, the motivated learners, do not have much use for formal teachers.

I would argue that the abacus has the added advantage (beyond kinesthetics) of forcing one to use placeholders in order to use it correctly. Someone like you, already curious, may not need that.

You raise a further great point: constructing one's notion of anything is critical in education, and we need more time for this. I got a lot of kids who are pretty handy with calculators, not so handy with numbers.

Great comments!

Dear earlsamuelson,

Pattern recognition is key--and lattice multiplication is one good way to get there.

We need to differentiate the classroom a bit so that those who have a more developed number sense are not wasting time doing exercises designed to help develop it. Elementary school was a nightmare for some of us--same crap over and over again.

Thanks for adding to the discussion!

Jenny said...

I have to agree - XKCD does rule.

As a first grade teacher I think the most important thing I do for my students in math is support their growing understanding of number sense. Without it stuff gets really rough by 3rd or 4th grade.

However, your final comment above gives me pause. It's possible I'm subjecting some kids to some pretty boring activities. I'm going to have to be more careful the year and in the future to ensure that I'm pushing kids forward as soon as they're ready while continuing to support those who need it.

doyle said...

Dear Jenny,

Are you kidding? Your students are blessed.

(Still, if you see some child who has gotten the number stuff down, let her doodle or read or finger paint or whatever it is that a bright 6-years-old does in school. I really hated 1st grade--in two different schools.)

Unknown said...

Alright Doyle, this is going to sound crazy, but I picked up an abacus last year for my eighth graders. Here's why:

Many of them learned algebra as a set of numerical properties. They memorized transitive, reflexive, additive, etc. However, these were things they memorized.

In other words, they didn't realize that they could manipulate numbers back and forth. While we did use other manipulatives (color-coded chips for digits versus variables) it clicked for some students when they used the abacus.

There is the very real sense that they learned math as a series of rules to follow and a process that always moves forward. Most of them had no idea that equations always have a user-controlled CTL Z.

Surprisingly, it was the abacus that helped students discover some of these conclusions.

doyle said...

Dear John,

Most of them had no idea that equations always have a user-controlled CTL Z.


Kudos for getting the abacus--kids who are innumerate (even those who do reasonably well in "math" courses) are easy prey for the nonsense spewed by people who know better.

David said...

You know, it seems foolish to me know to completely discount a tool that I haven't tried...(and which is really inexpensive). I'm happy to purchase some abacuses and see if they help.

The key is the ability of the students to play with the numbers. Like John says, they must recognize that they can control the process, rather than letting some process control them.