The handcuffs of NCLB

While taking a break from the work I was supposed to be doing recently, I found myself typing these words to a colleague:

It occurs to me that if the testing environment were not so toxic right now, we’d be ready for a real revolution in providing content. The technology is available to do a lot of open-source mathematics teaching; the ideas and community exist. But Larry and Stu aren’t really in a position where they feel they have the right/obligation/opportunity to move away from the published curriculum very far. If they toe the line and make modest progress on test scores (or none at all), they are covered. If they bust a move and see anything besides phenomenal gains in their first try, they have reason to be very, very worried. And furthermore they can’t count on collegial and administrative support along the way.

Those are the handcuffs that frustrate me. Without the constraints of NCLB, we wouldn’t need the big publishers much longer.

Within the constraints of NCLB, we get Khan Academy (which gets called ‘revolutionary’ but is not) and teachers are free to reform grading (SBG anyone?). But we don’t get meaningful curriculum change; least of all open-source curriculum.

Advertisement

Please raise your hand if you didn’t see this coming

I promise not to get worked up here.

But really. Are we surprised to learn that saying it doesn’t make it so? My six-year old is surprised by this fact, but adult educational policy makers?

Big education news this week (this from the New York Times):

The Obama administration has been facing a mounting clamor from state school officials to waive substantial parts of the law, which President Bush signed in 2002, especially its requirement that states bring 100 percent of students to proficiency in reading and math by 2014 or else face sanctions. In March, Mr. Duncan predicted that the law would classify 80,000 of the nation’s 100,000 public schools as failing this fall unless it was amended.

Once you have committed to the title No Child Left Behind (NO Child Left Behind), you have to commit to 100 percent proficiency, right? Not something that rounds to 100 percent. 100%.

Seriously, who didn’t see this coming ten years ago?

Please note that the passage above suggests 80,000 out of 100,000 public schools would be sanctioned this fall. But that doesn’t mean the remaining 20,000 are at 100%. No, it means they are still making adequate yearly progress towards 100% in 2014. Or that they have been until quite recently.

Which is still three years away.

TURD: Common Core graphic

I can’t quite decide whether this is a Truly Unfortunate Representation of Data. Help me out here.

The following is from an Educational Researcher article on the alignment between the math and English/language arts standards of various states and those of the Common Core State Standards (about which, more here).

The graphic (and several others like it) comes with the following disclaimer:

When reading these graphs, the representation of content emphasis is accurate at each column- by-row intersection, but the smoothing between rows and between columns is not meaningful because the data are nominal. (p. 107)

What this means is this, Because the data is categorical, we could really have put them in any order we like. As a consequence, any patterns (any patterns!) we see within each graph are simply artifacts of the order we chose. This smacks of TURD to me. But I stand ready to be convinced. Any takers?

Reference

Porter, A., McMaken, J., Hwang, J. & Yang, R. (2011). Common Core standards: The new U.S. intended curriculum. Educational researcher, 40, 103—116.

Integrated math-was that really necessary?

Bowen wrote in the comments recently:

Hey, speak for yourself, we dumped FOIL in our high school books. I guess that means we’re not mainstream!

I replied, with respect to CME, which is the FOIL-dumping curriculum to which he refers.

Is this news to you, Bowen? Is it bad news?

This got me thinking about high school curriculum-mainstream and non. Look back at the NSF-funded high school curricula of the 1990’s and the associated Math Wars. One major sticking point in the brouhaha was the idea that these programs were integrated. That is, they were not Algebra I, Geometry, Algebra II/Trig, PreCalculus/Statistics sequences. They were First Year, Second Year, etc. programs in which each year integrated algebra with geometry and statistics, etc.

And I have to ask, was that really necessary?

In hindsight, I’m not sure I really get the motivation for it. Wouldn’t it have been enough to improve high school pedagogy and content within each traditionally organized course? Did we have to try to blow up the whole structure at once?

I’m just asking.

More on complex fractions

Sean objected to my common core objections. He suggested the following context to motivate complex fractions as unit rates, as required by Common Core at grade 7 (and as explicitly proscribed by footnote at grade 6).

John is in a 10 mile walkathon for breast cancer. He looked at his watch when he started walking- it was 7:02. After a half mile, he saw that it was 7:17.

He suggests (and I concur) that students will likely use 1/2 mile per 15 minutes and get to 2 miles per hour (the desired unit rate), and then as teachers,

We write (1/2)/(1/4) on the board, and discuss its relationship to 1/2 per 1/4. This may be our major line of disagreement, as I don’t think this a terribly sophisticated jump. Assuming the students have some experience with slope and rate of change, this feels like fair game.

Indeed right here is the point of contention. It’s the transition from:

to:

As I wrote in the comments of my original rant,

[I]f I’m mathematically sophisticated and possess a graduate degree, then moving between (a) 1/2 mile per 1/4 hour and (b) (1/2)/(1/4) miles per hour is a very simple formal move justified by the multiplication algorithm for fractions, I suppose.

But walk into your average seventh grade classroom, make some orange juice from concentrate using the standard recipe (3 cans water: 1 can concentrate), pour precisely 1 cup of juice into a glass and ask, How much of this cup of juice is concentrate and how much is the water I added? My experience is that this is a challenging question for 12 and 13-year olds to make sense of. It’s really important at that grade level. It’s challenging to teach precisely because the relationship is so formally simple. But it doesn’t come naturally to lots of kids.

Answering the orange juice question requires shifting from ratio (cups water: cups juice) to unit rate (water per cup of juice), or from ratio to fraction. That’s hard, and it’s not even complex fractions.

The Common Core writers seem to want to move this from seventh grade to sixth grade. I’m OK with that; I wouldn’t have written it that way myself, but I have no major problem with it.

But they seem to think that we need to ramp it up in seventh grade and that the only way to do this is to do it with complex fractions. From the perspective of a middle school curriculum guy, I question whether it makes sense to do so. I don’t think it’s worth making kids do because it’s an unnatural representation. Furthermore, it’s machinery we don’t need. I don’t see a middle school problem that complex fraction unit rates will solve but conceptually simpler techniques will not. From my perspective as a college teacher, I don’t see it either. College Algebra? Calculus? Neither of these relies on complex fraction unit rates. It is conceptually much simpler to deal with (1/2 mile)/(1/4 hour) by either (1) equivalent fractions (multiply numerator and denominator by 4 to obtain (2 miles)/(1 hour)) or (2) division (divide 1/2 by 1/4, get 2).

I worked for a while with the best route to complex fraction unit rates that Sean suggested:

Assuming the students have some experience with slope and rate of change, this feels like fair game.

This makes sense; think of (1/2)/(1/4) as a slope, rise/run. But this leads me to want to divide. I don’t usually think of slope as 6/3 miles/hour; I divide and say the slope is 2 and the rate is 2 mph. So it doesn’t feel any better when the rise and run are fractions.

Sean concludes:

Obviously this isn’t perfect. But if complex fractions are a necessary component to a middle-school curriculum, where else do they land outside of unit rates and proportions?

Part-whole fractions. Area models. Here’s (1/2)/3:

And here’s 1/(2.5)

I’m all for complex fractions as they arise naturally in Sean’s walkathon problem-and they do arise naturally there. I am not for introducing a way of working with those complex fractions that is unnatural and has no special payoff.

I will gladly consider any and all suggestions for contexts and problems in which the complex-fraction-unit-rate gets me something that equivalence and division do not-and I’ll even accept examples from high school and Calculus. But I’m not optimistic that they’ll arise.

Update 6/7/11: I edited out a confusing statement about the ratios in the orange juice question. My question is about part-whole relationships, the explanation in the following paragraph alluded to part-part relationships. It’s fixed now.