severely fragmented knowledge

Looking through the SAT prep posts, I found this problem that, just 12 days ago, I couldn't begin to do.

If 20 percent of x equals 80 percent of y, which of the following expresses y in terms of x?

(A) y = 16% of x
(B) y = 25% of x
(C) y = 60% of x
(D) y = 100% of x
(E) y = 400% of x

The Official SAT Study Guide 2006
ISBN-13: 978-0874477184
p. 550

Today it seems simple and obvious.

This is a classic case of inflexible knowledge. I've practiced solving many, many literal equations over the past few years, a skill I learned and practiced in high school, too. I had no trouble doing it then, and I've had no trouble doing it now.

So, in theory, I 'know' how to do this problem.

And yet when I encountered a literal equation involving percent, I was mystified.

What fresh hell is this?


from cranberry:

Funny, I didn't need to set up a formal equation for this. If 20% of x = 80% of y, y must be four times as small as x. Thus, 25% of x.

That's the kind of thing years of doing bar models does for you.

Not that cranberry has spent years drawing bar models.

cumulative practice

I've been meaning to get a post up about this article for years now. I think it's incredibly important (relates to Direct Instruction, too).

No time to write now, but here's the abstract:

THE EFFECTS OF CUMULATIVE PRACTICE ON MATHEMATICS PROBLEM SOLVING (pdf file)
KRISTIN H. MAYFIELD AND PHILIP N. CHASE
JOURNAL OF APPLIED BEHAVIOR ANALYSIS
2002, 35, 105–123
NUMBER 2 (SUMMER 2002)

This study compared three different methods of teaching five basic algebra rules to college students. All methods used the same procedures to teach the rules and included four 50-question review sessions interspersed among the training of the individual rules. The differences among methods involved the kinds of practice provided during the four review sessions. Participants who received cumulative practice answered 50 questions covering a mix of the rules learned prior to each review session. Participants who received a simple review answered 50 questions on one previously trained rule. Participants who received extra practice answered 50 extra questions on the rule they had just learned. Tests administered after each review included new questions for applying each rule (application items) and problems that required novel combinations of the rules (problem-solving items). On the final test, the cumulative group outscored the other groups on application and problem-solving items. In addition, the cumulative group solved the problem-solving items significantly faster than the other groups. These results suggest that cumulative practice of component skills is an effective method of training problem solving.

Note: the effects of cumulative practice on problem solving.

Not "procedural fluency" or "automaticity" or "mastery" etc.

Problem solving.

The path to problem solving goes through a particular form of practice - cumulative practice - not through "do the problem 3 ways" (Trailblazers) or "explain how you got your answer."

a Math Trailblazers word problem

Lefty's post about the new research on word problems comes just two days after I found this word problem in a 5th grade Trailblazers homework pack:

Lee Yah and Manny are setting up tables for the 5th-grade family craft show. Each table is 200 cm long. Tables will line up along 2 sides of the room which measures 12 meters each. [could I possibly have copied this sentence correctly??] Each participant who wishes to sell crafts pays a $15 rental fee for 1 table or $25 for 2 tables. Mr. Moreno tells the students that all tables have been rented. Only 2 people rented two tables. How much money will the school make on the craft show?

The first thought that sprang to mind when I read through this problem was: no child can possibly learn math from this.

I myself find the thing close to inscrutable. (Do the same people who write instruction manuals write word problems for Trailblazers?)

fyi: one of ktm's frequent commenters sent me this observation in an email just last week:

But what I have long worried about is that for some reason, the skills taught in reform math programs do not *transfer* well. I have never seen anyone talk about that, but I've seen it time and again. Their learning seems to be really confined to the particular context/problem they learned it in.

Query: Does teaching new concepts in a "real world context" have the inadvertant effect of obstructing the transfer of that learning to other contexts? Do kids who learn things in the abstract have an easier time of it?

I get $170.

Problems with "real world" math problems

I've been complaining for years that all the hands-on, "real world" math activities that dominate Reform programs shortchange kids on the autistic spectrum. They confuse the many who have language delays and gaps in worldly knowledge, and are too full of distractions for those who require streamlined, structured learning environments.

Now a new study by Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State, reported in today's NYTimes, suggests that this kind of hands-on math is bad for everyone.

Quoting the Times: "The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems."

As the Times also remarks: "Dr. Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler did something relatively rare in education research: they performed a randomized, controlled experiment."

See One, Do One, Teach One

The title refers to how medical interns learn procedures. But it is also true elsewhere. From the Public Education Network Weekly Newsblast, this short article covers two studies:

MOMMY CAN YOU HEAR ME? CAN YOU HELP ME LEARN?
The goal of a new study from Vanderbilt University was to examine whether explaining to another person improves learning and transfer. In the study, four- and five-year olds solved multiple classification problems, received accuracy feedback and were prompted to explain the correct solutions to their moms, to themselves or to repeat the solutions.

The study found that generating explanations improved problem-solving accuracy following the test, while explaining things to the mother led to the greatest problem-solving transfer. This indicates that explanation prompts can facilitate transfers in children as young as five years and reveals that it matters if the mother is listening. Even though it is possible that prompting children could be a substitute for the positive influences of a listener, there is reason to suspect that explaining to another person improves learning.

People often produce more detailed and explicit explanations and justify their ideas more when they are doing so for other people rather than just for themselves. And young people adjust their speech based on the age of the listener. It follows that explaining to others may increase motivation and also support more complete and explicit knowledge. This improved knowledge could be more easily transferred to new situations and problems.

The Vanderbilt study’s sentiments seem to be echoed in a new piece in Teacher Magazine written by Kathie Marshall (second link). Marshall begins by noting that a great deal of research has been conducted on the importance of student discussion and its prevalence in class. However, research from Martin Nystrand finds that eighth graders spend an average of 50 seconds per class in sustained conversation, with ninth graders spending only 30 seconds. So Marshall set out to see if discussion could help improve struggling students.

After four weeks, Marshall was amazed at the results of this strategy. Many ‘C’ and ‘D’ students were suddenly performing at the top of the class and were highly engaged in their work. One student wrote in her notebooks that "I like this class because in our other classes, we get in trouble if we want to talk about what we are learning."

Learning from Explaining: Does it matter if mom's listening?

Best Practices: Getting to Comprehension in the Classroom by Kathie Marshall Feb 19, 2008

inflexible knowledge in action

C. called home after school to say that the math test was "pretty easy."

"But, oh mom!" he said. "It didn't have the problems we did!"

"What problems?"

"It didn't have population problems or problems about the price of something."

"It didn't?"

"No!"

"What kind of percent problems did it have?"

"Problems about markups and discounts."


Inflexible Knowledge: The First Step to Expertise

The Executive Brain

Research has shown that the capacity for generalization in problem solving is limited even in neurologically healthy people....People tend to learn by acquiring situation-specific mental templates.

The Executive Brain: Frontal Lobes and the Civilized Mind
by Elkhonon Goldberg


For me, The Executive Brain was a page-turner.

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I'm repeating myself! (you may need to hit refresh a couple of times to make the page appear)