kitchen table math, the sequel: get the MESAG, part 2

## Monday, September 3, 2012

### get the MESAG, part 2

Back from the Open, so summer is officially over, as opposed to emotionally over, which it was the instant Chris set foot inside his dorm.

Hate the empty nest! hate! hate! hate!

Anyway, getting back to my interrupted post on precision teaching, when you learn something to fluency, you "get the MESAG":

Is fluency the magic that makes inflexible knowledge flexible?

Anonymous said...

You said "What I do know is that precision teachers believe that fluency renders skills transferable to new contexts and problems."

I'd like to see some evidence for that, as it goes against my intuitions about how the term "fluency" has been used on this blog before. Perhaps I just need a clearer definition of fluency.

SteveH said...

Is there some official or common definition of fluency? I've never been able to find any definitions for understanding, problem solving, and critical thinking. How about "deep understanding"?

Can you have the fluent ability to solve problems without understanding? Some think that you can have fluency with little understanding. They call that rote. I've always claimed that there is linkage. But is fluency defined as just speed? Are there different levels and meanings for speed? Is Jeopardy speed related to fluency?

My view is that fluency is more than speed. Does fluency allow for flexible knowledge? I would say that fluency is defined by flexibility.

When you learn about the D=RT formula, it's easy to solve for distance if you are given a rate and a time. What if you are given values with different units? What if you are given distance and time? What if you are given two legs and two different rates and need to find your average rate? What if there are two trains heading in different directions? What does fluency mean? It has to mean more than speed. It has to mean more than pattern matching to problems you have seen before. When you do "enough" problems of different types, then rote pattern matching evolves into flexibility and understanding. When students never get past the rote level, many educators assume that there has to be an alternate top-down understanding approach that will work. It doesn't. The problem is that they give up to soon. They never ensure fluency at each step of the way. As the years go by, students find it harder and harder to achieve fluency. In its place, many educators talk about fluency of general top-down Polya-type problem solving methods. They are no replacement.

If you are fluent in the use of the D=RT equation and all sorts of variations of it's applications, then, of course, that means that you have more ability to apply the equation to new situations. General problem solving methods are no substitue for a bottom-up, mastery approach to individual mathematical techniques.

I would say that fluency IS flexibility. Look at any proper math textbook problem set. They start with simple applications and move towards more complicated usages. The whole goal is fluency or flexibility.

Unfortunately, CCSS uses the word "fluency" very little. I think I once counted it about five times in the CCSS Math document. It seems that they are treating fluency and understanding as two separate things, with fluency being used for things like the times table and basic algorithms. They talk of understanding first and THEN fluency. That's backwards unless you are talking about conceptual understanding. Speed is one indicator of fluency, but it is much more than that.

In general, the vagueness of terms has always been a problem (cover?) in the world of education. I've found that educators argue with generalities to get parents to go away so that they can decide on all of the important details. When my son was in fifth grade, we had a parent/teacher meeting about Everyday Math and the discussion was all about how great "balance" is. Nothing changed.

Catherine Johnson said...

gasstation - I've got to get my syllabus together, but as soon as I get time I'll rustle up some examples of fluency leading to application. The idea that fluency produces generalization is a core element of the definition.

Catherine Johnson said...

Is there some official or common definition of fluency?

Yes!

Fluency means speed and accuracy, which is the Kumon definition.

I think of fluency as including:

* speed
* accuracy
* nonconscious performance

I've mentioned several times that I've been working on a basal ganglia project ... and, as it turns out, precision teaching is all about the basal ganglia. Fluency happens in the basal ganglia, and basal ganglia performance is essentially nonconscious.

The consciousness issue has been, I think, confusing for many of us (for me, at least) because 'nonconscious' performance doesn't mean that you have to be oblivious to what you're doing.

You ***can*** be oblivious -- you can 'do it in your sleep' -- but if you want to pay attention to what you're doing when you tie your shoes, for instance, you can.

Of course, paying conscious attention to a fluent, nonconscious skill very often (always?) interferes with performance. (Choke is the book to read on this.)

Haven't read all of Steve's comment closely, but I don't think you can be 'fluent' in problem-solving.

You definitely can't be fluent in WRITING ABOUT THE BASAL GANGLIA. I can tell you that for sure.

You can be, and you should be, fluent in the tool skills that allow you to solve problems or WRITE ABOUT THE BASAL GANGLIA.

The Morningside people give all their students fluency training in sentence combining!

I think that is bloody brilliant.

Catherine Johnson said...

fluency is more than speed

Fluency as precision teaching people define it is speed, accuracy (close to 100% accuracy), and nonconscious performance.

I'll find my favorite short definitions from the literature...

Anonymous said...

Catherine, your definition of fluency in terms of speed, accuracy, and nonconscious performance agrees with my understanding of the term.

With that definition, I don't accept that fluency renders skills transferable.

With SteveH's definition (which seems to me to be to "understanding" not "fluency"), transference seems more plausible.

I think SteveH's point is that fluency (with your definition) is essential to building understanding, which I think is at least partly true. Fluency reduces the amount of working memory and attention needed to basic tasks, allowing more attention to higher-level skills, which are the core of understanding.

If he can do long division in his sleep, he can do long division inside Penn Station.

This is true.

I've had two MRIs in the last eight years, the kind where you lie in a tube for 20-30 minutes while you're bombarded with jackhammer noises. During the first one, they had me wear headphones with music that wasn't quite loud enough to drown out the jackhammers, and I nearly fell asleep.

The second one was last year, and I fell back on my old standby of reciting the squares, in order, in my head. You know, 1, 4, 9, ... I can generally get up to somewhere in the neighborhood of 1600 before I fall asleep or I finish the excruciatingly boring task that prompted me to do the squares in the first place.

Yes, you can recite squares in your head while the MRI magnets are pounding away.

SteveH said...

"Fluency as precision teaching people define it is speed, accuracy (close to 100% accuracy), and nonconscious performance."

That's a very limited definition and not what I see when people do homework sets. It's not just speed and accuracy they are developing. As I tried to show with the D=RT discussion, you develop something other than speed and accuracy. You develop the ability to apply your understanding of governing equations to new situations. This is not an unconscious process. Separating proper fluency and understanding is not possible.

If you were to become fluent in the use of the Bernoulli equation for fluid flow in pipes, solving problems would not necessarily be an unconscious process. Fluency is not only for problem variations you have seen before. Many educators like to separate fluency and understanding because they want to emphasize alternate (usually top-down) approaches where understanding evolves separately from algorithmic fluency. This is backwards. Fluency is more than just freeing up your mind to think about other things.

Can you define fluency up to a hard cut-off that is followed by an understanding phase? I don't think that's what happens. There are multiple levels of understanding. Can you have fluency with little understanding; what some call a rote ability? That's not what I call fluency. Perhaps you can be fluent in long division but not see much understanding. But what does it mean to be fluent in fractions or percents? Fluency implies different things for different skills, but fluency has to be more than rote at every level of understanding. Fluency has to be defined as flexible. If you aren't flexible, you aren't fluent. Perhaps you can see all problem variations for some skills, like long division, but that's not true for most things you have to learn in math.

With that definition, I don't accept that fluency renders skills transferable.

GWP,the PT people have a lot of data to substantiate this claim, at least in certain areas (I paid more attention to the language and reading data than to math however).

Generalization may be a better term than "transfer" although certainly "transfer" is what is happening in specific skills taught that can be readily measured and monitoreed. I could give a few examples, but if you're really interested in the science behind it you can research "contingency adduction" and work by Layng, Johnson, Kubina, Andronis (I think) and many others.

For complex higher-level math and science, as well as reading and written language, fluency and understanding are obviously interconnected and difficult (or maybe impossible) to measure precisely or discretely, AFAIK.

However, building up deficient component skills (what PT people call "tool skills")often precipitates a "quantum leap" in proficiency (including understanding) in higher-level complex tasks in language and mathematics. I am not aware of any research into accurate measurement of ALL the variables involved in such tasks. The whole is greater than the sum of its parts; however, a missing part can significantly impair the whole.