kitchen table math, the sequel: I am uncertain

Sunday, March 4, 2007

I am uncertain

Back in 5th grade, on his TONYSS test, Christopher scored exceptionally well on "uncertainty."

I had no idea what "uncertainty" could possibly mean.

He was a mess on measurement, a near-genius on uncertainty.

No one at the school could tell me what either scale measured.

Now I know.

"Uncertainty" means "probability and statistics," which, in NY state, means a) counting stuff and b) constructing bar graphs to show the totals.

to wit:

Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.

This must be one reason for the crowding out of algebra by probability-and-statistics.

Probability-and-statistics means there's no one right answer!

Needless to say Fordham takes a dim view:

The classification scheme of the performance indicators, according to the Key Ideas, compromises the quality of New York’s standards. Too much emphasis is placed on patterns, probability, and data analysis. The sixth Key Idea, “Uncertainty,”... is misleading and a poor choice of category for performance indicators. It mistakenly associates ambiguities inherent in choosing mathematical models for “everyday situations” with mathematics itself.

17 comments:

Anonymous said...

I read through some of the Math Panel transcripts and they argue about the crowded curriculum. It's a mess. The justification for wanting to include even MORE is that they are going to be very efficient (what are they waiting for?) in the lower grades and that will make room in the upper grades for more topics.

Wade said...

It's also confusing an important concept in statistics and the sciences. Uncertainty means something specific, and is tied up with random and systematic error. Real uncertainty (or error analysis) is very persnickety and calculus based, and my junior and senior chem majors dread it. It certainly isn't what junior high school kids are doing. Instead, they are introducing basic statistics and giving it a "high falutin'" name.

Tracy said...

Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.

But the whole point of applying maths to everyday situations is to get some more exactness into it.

Of course sometimes the exactness may involve knowing how much you are not being exact. But you don't need uncertainty to get to that. For example, many theories in monetary economics talk about the effect of money in the economy using mathematical terminology. When you try to test these theories you realise the definition of money is very inexact (Does it include paper money? Credit cards? On-call accounts? What do I mean by on-call accounts?)

Anonymous said...

Instead, they are introducing basic statistics and giving it a "high falutin'" name.

It's "show" math. "Look, parents, what we are doing!"

Barry Garelick said...

In order to appreciate the use of "uncertainty", one has to gain mastery over the math that DOES give you one right answer. E.g., to understand "line of best fit", you have to know what slope and linear equations are all about. The fuzzies hold in disdain word problems in algebra that don't represent the messiness of real life. So they rail against "work problems" (e.g., John can shovel the walk in 30 minutes and his brother in 20 minutes; how long does it take for them to do the job together?) not recognizing that such problems represent a form of solution that is generalizable to many other problems one comes across in the sciences. They concentrate on the specific messy problem without going over the underlying theory that allows one to generalize a solution.

Instructivist said...

"Too much emphasis is placed on patterns, probability, and data analysis."

That overemphasis is a pattern I've also noticed and wondered about. What thinking or motivation could be driving this overemphasis?

I see the lust for "uncertainty" as an expression of the fuzzies' aversion to exactness. I would have thought that approximations like rounding and irrational numbers like pi should be enough inexcatness to satisfy this lust.

rightwingprof said...

"In order to appreciate the use of "uncertainty", one has to gain mastery over the math that DOES give you one right answer. E.g., to understand "line of best fit", you have to know what slope and linear equations are all about."

Exactly. And "uncertainty" isn't "fuzziness" or "estimating an answer" or "there could be many answers and we like yours just fine!"

Catherine Johnson said...

Hi, Wade!

yes, thanks so much for your observation (I'm going to try to get it posted up front).

Having taken one statistics course in college (not calculus based) I simply didn't manage to make the connection between what schools are calling "probability" and what I was taught about "uncertainty."

Catherine Johnson said...

But the whole point of applying maths to everyday situations is to get some more exactness into it.

EXACTLY!

That's roughly (as I understand it, at least) the idea of "uncertainty": the hope is to make a reasoned determination that an event was unlikely to have occurred by chance.

"Uncertainty" is about the reduction of uncertainty.

(Have I said that right?)

Catherine Johnson said...

For example, many theories in monetary economics talk about the effect of money in the economy using mathematical terminology. When you try to test these theories you realise the definition of money is very inexact (Does it include paper money? Credit cards? On-call accounts? What do I mean by on-call accounts?

Yes!

Again, I can speak only as a very interested and very undereducated "bystander," but this is how I've always understood the field of statistics, probability, etc.

Catherine Johnson said...

In order to appreciate the use of "uncertainty", one has to gain mastery over the math that DOES give you one right answer. E.g., to understand "line of best fit", you have to know what slope and linear equations are all about.

yes!

this thread is making me very happy...

Catherine Johnson said...

That overemphasis is a pattern I've also noticed and wondered about. What thinking or motivation could be driving this overemphasis?

I see the lust for "uncertainty" as an expression of the fuzzies' aversion to exactness.


I've had the same question for quite some time now.

I do think it's a good idea to teach students statistics (unless everyone here wants to persuade me it's a bad idea!).

But why do we see 4-red-marbles-3-blue-marbles problems year after year after year???

It hadn't occurred to me that such problems satisfy the need to create problems without right answers.

Leaving aside, of course, the fact that 4-reds-3-blues has a right answer.

Catherine Johnson said...

That overemphasis is a pattern I've also noticed and wondered about. What thinking or motivation could be driving this overemphasis?

I see the lust for "uncertainty" as an expression of the fuzzies' aversion to exactness.


I've had the same question for quite some time now.

I do think it's a good idea to teach students statistics (unless everyone here wants to persuade me it's a bad idea!).

But why do we see 4-red-marbles-3-blue-marbles problems year after year after year???

It hadn't occurred to me that such problems satisfy the need to create problems without right answers.

Leaving aside, of course, the fact that 4-reds-3-blues has a right answer.

rightwingprof said...

""Uncertainty" is about the reduction of uncertainty."

Yes. Say we have two groups of students we have tracked from grades 1-6. One group went through a traditional math program, and the other went through a fuzzy program. When we look at the math scores for the two groups on the state exam, we see that the mean score for the traditional group is higher than that of the fuzzy group.

Here's the uncertainty: 1.) Is the difference the result of random variation, or not? 2.) Is the difference between the two means large enough to be significant?

We use statistics to answer those two questions, though the answer itself is uncertain. We can never say we KNOW anything from statistics, only that we can conclude to a specific probability (95% or 99% usually) that the scores are the result of the curriculum (or not).

Tracy said...

That's roughly (as I understand it, at least) the idea of "uncertainty": the hope is to make a reasoned determination that an event was unlikely to have occurred by chance.

"Uncertainty" is about the reduction of uncertainty.


Not necessarily. Sometimes it's about measuring uncertainty (a friend spent his PhD proving that metoerite streams are random) or even increasing it. A casino does not want their customers to be reducing their uncertainty, so they aim to make roulette tables/slot machines/etc as truly random as possible.

Catherine Johnson said...

We use statistics to answer those two questions, though the answer itself is uncertain. We can never say we KNOW anything from statistics, only that we can conclude to a specific probability (95% or 99% usually) that the scores are the result of the curriculum (or not).

I was taught this.


Sometimes it's about measuring uncertainty (a friend spent his PhD proving that metoerite streams are random) or even increasing it.

Is that definitely a separate thing?

To me, measuring uncertainty seems to make uncertainty less uncertain (now there's a sentence!)

At a psychological level, that's probably (probably!) true - this is why people have amnios, go for genetic counseling, etc. (I HAVE LIVED THIS TALE, BELIEVE ME)

In the case where you're trying to increase uncertainty because you own a casino, aren't you also trying to raise the certainty that you'll make a profit??

Or is that wrong....

Tracy said...

To me, measuring uncertainty seems to make uncertainty less uncertain (now there's a sentence!)

I am afraid the subtleties in this sentence have lost me. Perhaps if you could cast it mathematically?

At a psychological level, that's probably (probably!) true - this is why people have amnios, go for genetic counseling, etc.

And some people choose not to. My grandma occupied years by fretting about whether she had cancer while refusing to actually go to the doctor and get checked out. (Eventually she died of a massive unexpected heart attack in her seventies).

In the case where you're trying to increase uncertainty because you own a casino, aren't you also trying to raise the certainty that you'll make a profit??

That's why I talked about what the casino wants for its customers, not what the casino wants for itself.