Wrongness

The topic of being wrong pops up more and more frequently in public discourse these days. Author Chuck Klosterman, maintains we are probably wrong about everything we think we know, including and maybe especially gravity. Meanwhile, we are chided for being “ant-science” if we disagree with the consensus of scientists. In a famous Last Week Tonight spot, Bill Nye (the Science Guy) leads a climate change “debate” that was no more than Bill with 96 white-coated people representing the 97% of the scientific consensus against 3 other people representing the 3% of the science community refusing to join the bandwagon. Case closed, apparently.

We all “know” that Republicans are the anti-science party, right? Except, according to Neil Degrasse Tyson, there is plenty of anti-science on the Liberal side of the aisle as well. Steven Novella, MD, a contributor to Neuroligica Blog, supports Dr. Tyson’s assertions with some survey results, concluding, “My synthesis of all this information, which is admittedly incomplete, is that people tend to be anti-science whenever science confronts their ideology.”

Dr. Novella elaborates,

I think it is more meaningful to understand these issues by breaking them down to specific ideologies and how they influence acceptance or rejection of science. Conservatives tend to value freedom, the sanctity of life, and the free market and they distrust government. Liberals value nature and the environment and distrust corporations. Individual issues are complicated because they can cut across multiple ideologies. In terms of the question of who is more anti-science, my approach is this – you don’t get credit for being pro science for accepting an issue that is compatible with your ideology (bold added). Liberals acceptance of manmade global warming does not mean they are necessarily pro science, because this issue is right in line with their ideology (pro nature, anti corporate). Conservatives don’t get credit for being pro nuclear for the same reason. Evidence for being pro science is when you accept a scientific consensus that conflicts with your ideology. You have to demonstrate that science comes before your ideology, (bold added).

The thing is the 3% of scientists who disagree with the 97% are not wrong simply because they are outnumbered, as Bill Nye implied. Science is not a majority-rules proposition. Throughout history, there have been scientists who have disagreed with mainstream science. Some suffered, at worst, outright scorn and ridicule, or at best, indifference, only to be found to have been right all along. One big reason why accusations of being “anti-science” carry no weight with either camp is because everybody knows that settled science is settled only until a scientist unsettles it.

“Anti-science” is the new heresy. There is nothing wrong with disagreeing with settled science. The problem is when disagree-ers (of any stripe) have no basis for the disagreement except ideology. That’s a problem that seriously impedes useful discourse on any issue.

Missing Key to Understanding Place Value

I write a lot about place value. Place value (along with zero) may arguably be the most important math concept because it underlies every single calculation we do. Yet teachers often do not teach place value well. Teachers (and most curriculum) are satisfied with a very superficial understanding of this essential concept. If a child can identify the place name of a given digit or put a digit in a given place, most teachers deem the child to have a good understanding of place value. Place value is so much more.

Groups of Ten

Place value is all about making groups of ten. Well, yeah, the reader might say. Tell me something I don’t know. The key to understanding place value is the realization that each succeeding place represents a group of ten of the preceding place. Duh. Stay with me here. The curriculum and instruction alludes to this key, but rarely makes it explicit. Most textbooks have replaced “borrowing and carrying” with “regrouping,” and this was a positive step, but students still take a mechanical view. They still borrow and carry as they move leftwards through an addition or subtraction problem without realizing that they are actually making or breaking a group of ten at each successive place. For example, if they carry a one from the tens place to the hundreds place, they mechanically add that one to the other digits in the place without realizing that the carried one represents making a group of ten. In fact, most students will say, (correctly on a superficial level), that they made a group of 100 because they put the “1” at the top of the column named “100s place.”

Place value is all about making groups of ten. Subtraction is all about breaking groups of ten into loose ones and dumping them with the other loose ones. Every place except the loose ones is a group of ten something. Teachers tell students that each succeeding place is larger by a magnitude of ten, but somehow children fail to grasp the significance of this fact. The reason the standard addition algorithm works is because you are gathering up groups of ten at every place. Likewise, the reason the standard subtraction algorithm works is because you are breaking a group of ten at every place.

Students betray this lack of deeper understanding when they express surprise that given the number 437, that an equally correct answer to the question “How many tens?” is 43. They are also surprised to learn that when we say 2 tens and 5 ones equals 25, what we really mean is 2 tens and 5 ones equals 25 ones.

A better way to express it is “2 groups of ten and 5 loose (not in a group) ones equals 25 loose ones.” Therefore, I spend a lot of time having students expand large numbers in a variety of ways.

Methods of Expansion

Expansion basically means counting numbers of groups. There are several ways to express this accounting. Given the number 47,396:

Standard Methods:

Place Value Names: 4 ten thousands, 7 thousands, 3 hundreds, 9 tens, 6 loose ones

Multiplication: (4 x 10,000) + (7 x 1000) + (3 x 100) + (9 x 10) + (6 x 1)

Exponents: (4 x 10^4) + (7 x 10^3) + (3 x 10^2) + (9 x 10^1) + (6 x 10^0)

Notice that using exponents displays the idea that each succeeding place is a group of ten, however, most teachers do not make this understanding explicit. Most students just view, for example, the number 10000 or 104 as merely another way of expressing the place value name “ten thousands.”

I give my students practice with alternative expansions.

Alternative Expansion

47, 396 = _______ thousands, ________tens, _____ ones

47, 396 = _______ ten thousands, ________hundreds, _____ ones

47, 396 = _______ tens, _____ ones

And of course, we can repeat this exercise with multiplicative expansion and exponential expansion. This sort of practice has the side effect of helping students later understand rounding to a given place. I am also very picky about counting and zeroes. 0 is a real counting number, and I expect students to show that they know that 102 has 0 tens, or (0 x 10) or (0 x 10^1).

Place Value in Later Mathematics

This sort of foundational learning of place value pays dividends in later mathematics. To give just a couple examples:

Bases: Each succeeding place is a group of the given base. This understanding gives logic to “borrowing and carrying” in other bases besides base ten.

Polynomial expressions: Quadratic and other equations of the form Ax^n + Bx^n-1 + …Gx^1 + Hx^0 are essentially equations expressed in base x. Students will find that working in other bases is greatly simplified if they exponentially expand the number and replace the base with x.

Polynomial (and by extension, synthetic division: When students learn to divide equations such as Ax^3 + Cx^1 + D by say, x + 1, they must remember to insert the missing term, 0x^2. Students do learn to replace the missing term in a mechanical way. However, if they have regularly understood zero as a real counting number and included the zero term in their elementary expansions, it seems obvious to them that of course they must have the zero term if they expect to successfully complete the division.

More attention to a deep understanding of place value in the early years would make much of later mathematics less mechanical and more intuitively comprehensible, thus actually saving instruction time and allowing teachers to teach more math.