Preconceived notions about place value

(Cross-posted at Out in Left Field--with some great comments)

One thing that struck me about the math talks given at this weekend's New England Conference on the Gifted and Talented was the emphasis on manipulatives and the concerns about whether children understand place value. Are these the most appropriate things to be focusing on when it comes to students who are gifted in math? The mathematically gifted kids I know grasp place value and other aspects of arithmetic with only minimal exposure to manipulatives, and quickly advance to higher levels of abstraction by the time they hit first or second grade. But the education establishment seems bent on convincing itself that children--however gifted--don't understand place value.

Why would you want to convince yourself of this? Because it gives you an excuse not to teach the standard algorithms of arithmetic. If children don't understand place value, then they can't understand borrowing and carrying (regrouping), let alone column multiplication and long division. And unless they understand how these procedures work from the get-go, educators claim (though mathematicians disagree), using them will permanently harm their mathematical development.

So, given how nice it would be not to feel any pressure to teach the standard algorithms (because, let's admit it, they are rather a pain to teach), wouldn't it nice to convince ourselves that our elementary school students, however gifted in math, don't understand place value?

But how do you convince yourself of this? As that ground-breaking math education theorist Constance Kamii has shown, it's child's play. All you have to do is ask a child the right sort of ill-formed question. Here's how it works:

1. Show the child a number like this:

27

2. Place your finger on the left-most digit and ask the child what number it is.

3. When the child answers "two" rather than "twenty," immediately conclude that he or she doesn't understand place value.

4. Banish from your mind any suspicion that a child who can read "27" as "twenty-seven" might simultaneously (a) know that the "2" in "27" is what contributes to twenty-seven the value of twenty and (b) be assuming that you were asking about "2" as a number rather than about "2" as a digit.

for Paula

Trawling the web last night, I found a set of PowerPoint slides (pdf file) Thomas H. Parker used in two presentations to the Core Knowledge National Conference on March 5, 2004.

Here's what he had to say about place value, and in this color, too:

Place value causes MANY, MANY problems all through elementary school

Parker also has this to say:

This idea of using position to encode value is a secret code. Because this is second nature to adults, it is easy to overlook the fact that it is difficult to learn. It requires explicit teaching and must be repeatedly worked on. Place value ideas occur and present difficulties in topic after topic in elementary school.

I haven't read the slides yet, but they look helpful, and include place value exercises.

source:
Sequencing in Elementary Mathematics
Thomas H. Paker

place value makes the "short list"

cross-posted to Beyond TERC:

terrific post on place value:

Place value is one of those things non-mathematically trained grownups tend to take for granted (at least, I did).

The Singapore Math series teaches place value year after year. Singapore Math has a "true" spiral curriculum in that kids learn to mastery in year one, then study the same topic in more depth in year two and master that material, too, then study the same topic in still greater depth in year three and, again, master the material. Place value is one of the spiraled topics.

I didn't quite understand this, even though Christopher's brilliant 5th grade teacher (I don't use the term "brilliant" lightly) told me how important the topic was. She said she'd asked her friend, who had a Ph.D. in math education, what were the most essential & fundamental topics for K-5 kids to master.

He said "place value."

I wish I could remember what Ed said about it the other day. We were talking about some "guess and check" problem-solving situation that was supposed to be a model of higher order thinking.

Ed said, "The way these kids are solving the problem shows they don't understand place value, and if they don't understand place value they don't have conceptual understanding."

One of the things that would be SOOOO helpful to parents like me (to most parents, that is) would be to have a list of the CORE topics you MUST make sure your child knows.

For instance, the other day Vicky said that decimals aren't as important as fractions.

I sort-of knew that already, but only because I've spent 2 1/2 years of my life immersed in K-12 math & math education. But, otoh, I didn't know it with conviction.

Most parents can't teach math on the side; even parents who have the capability to teach math on the side are going to find that their kids won't cooperate past the age of 10. (You can still teach a middle school child on the side - I've done it - but the amount of time you have to spend wrangling with them to get their attention steeply reduces the amount of time on task.)

What we need is a short list of THE essential skills our kids MUST have.

My list, so far, is:

• fractions

• place value

• long division
  

• measurement
  

from Lynn:

• automatic recall of basic facts

from independent george:

• order of operations
  

• properties of arithmetic

from instructivist:

• In addition to total mastery of math facts, I would put equivalent fractions on top of the list. Equivalent fractions directly leads to an understanding of proportions, percentages and all the other good stuff (scaling, unit rates...)

from Mr. Person:

I've had a list for a long time of core concepts/skills that are emphasized at each grade level.

1: Place Value
2: Addition/Subtraction Facts
3: Multiplication/Division Facts
4: All operations with whole numbers
5: All operations with fractions
6: Ratios and proportions

from Susan J:

Estimation.

Common sense. (Such as if you subtract a positive number the result should be smaller.)

[question: do the reform math programs do a decent job with estimation and common sense? do we know?]


update: it strikes me that the list already exists. It's the TOC for the Primary Mathematics series.

I'll pull it & post.