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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."
11 comments:
Good grief.
I already didn't write a post about this study back in 2007.
I did not get the impression from reading the CCSSI math draft that cumulative practice is a high priority.
In fact, I think anything that might reenforce the fundamental life fact that there's a hierarchy of individual abilities in math will be frowned upon in practice.
If everyone did cumulative practice, some would outscore others in problem solving and that's no longer acceptable.
A primary political benefit to pushing Balanced Literacy is the fact that it obscures the fact that children from certain backgrounds just simply need more training and exposure to crack the phonetic code and develop phonemic awareness.
Apparently part of Klein's support for BL for NYC and disdain for Success for All was his belief that it is unacceptable to have different types of reading instruction in the public school system.
Real differences in needed scaffolding get ignored in favor of the "it works for everyone" emphasis.
Does that sound like the rationale for the inquiry learning math programs to anyone?
In describing NYC's Balanced Literacy program, one of the proponents described how it supported everyone in the class exercising "critical thinking" skills by talking about the characters in their respective books.
Wow! That's high level thinking skills all right.
The Asians must be quite paranoid that we are pushing national standards for an inclusive classroom.
Balanced Literacy and Inquiry Math- they may not be effective models but everyone can use them together.
Now it's starting to all make sense.
Saw this morning that the new ESEA blueprint will be pushing even more mainstreaming in the general ed classroom of
-special ed students and
-English language learners
It makes sense then that Common Core would have to allow for all inclusive instruction models.
So the content language is the lure to make the standards appear more substantive than what most states have now.
The skills language and the vagueness of "understand" make more sense now.
I do not think we are going to like the actual implementation of Common Core at all. In fact, given that disparate impact college prep analysis (NYT article) Allison linked to, will any instructional model or textbook that is not all-inclusive be argued to be discriminatory?
If so, Common Core will mean that the all inclusive model will become the mandated national norm in K-12 classrooms.
Oh my. I hope that's not where all these federal programs are leading to.
well yes & no
here in NY, we will have mandated "Response to Intervention" in 2010
RTI requires that struggling students be taught in ratios of 3:1 to 5:1
Now maybe that will be done as push-in rather than pull-out....
but still: it's not exactly full inclusion ---- not as I've understood full inclusion, at any rate
Even though it is questionable whether the disparate impact analysis described above and being pushed by Justice is constitutional, schools and school districts always engage in overkill once they start worrying they might get in trouble.
Think of all the principals that honestly believed that no one was to speak on any religious topic, voluntarily or not, while on school grounds.
It does seem likely we are headed toward a period where the heterogeneous classroom will be the only safe way to avoid constitutional scrutiny. That clearly impacts what can go on in the classroom. Imagine the range of math problems that would be needed for cumulative review by 8th grade in such a classroom.
It looks like we will be left then with ineffective and inefficient instructional models.
Also as Catherine noted on Jay Greene's blog and we've talked about here, this creates the need for more staff and that's unaffordable in this new era of finite public dollars.
Isn't it ironic that the most effective practices are cheaper to boot?
Is that part of the reason they must be shunned?
I don't think we'd be facing this perfect storm of classroom composition, terrible curriculum and terrible instructional methods if the achievement (test scores) gap didn't break down along racial/ethnic lines. The diversity mantra cannot be challenged, even by facts, so something must enable the pretense that all are equally able and motivated.
Why anyone should be surprised at the racial/ethnic differences, I don't know. There is a continual stream of just-arrived, non-English-speaking immigrants with unskilled, uneducated parents (largely Hispanic) on one hand and large numbers of kids (largely Black, but increasing among whites) who are products of several generations of very young, never-married mothers and absent fathers and who are pretty much raising themselves in a gang culture of the streets on the other hand. In both of these groups, there is too-often explicit pressure to view academic success as a betrayal of their racial/ethnic identity. Schools cannot adequately worship diversity and explicitly demand the behaviors that enable academic success because to do so would imply that some cultural behaviors are better than others.
Even poor Asian immigrants have succeeded because they typically come from intact families and their culture typically demands max effort and academic success.
I have never known a young teenager capable of meeting the demands of good parenthood; I doubt that it's even possible in a modern culture. The only way the kids at the bottom have a chance is for the schools to provide a content-rich curriculum and explicit instruction, to teach the habits and behaviors that enable academic and work success and to demand their use. However, to do this violates the diversity mantra and all of the most cherished ed-school philosophies about kids, teaching and learning.
Isn't it ironic that the most effective practices are cheaper to boot?
Is that part of the reason they must be shunned?
I have come to think that schools persistently choose less effective & efficient methods & curricula precisely because those methods are cheaper & faster.
I don't think conscious intent is involved; I think that throughout the system existing incentives are bad --- and I believe that incentives often (or usually?) operate at the unconscious level.
That's what makes them so hard to combat.
At the moment I'm thinking that in order to understand the way in which public schools have grown & evolved, half your analysis has to be B.F. Skinner (incentives shaping behavior) and the other half has to be Sigmund Freud (defense mechanisms rationalizing what is as good).
As to Freud, it's amazing to me how often you see what to us is a bad situation (5 year olds 'taking risks' while attempting to read 3-letter CVC words) as good ('children are risk takers and decision makers'!)
I don't think we'd be facing this perfect storm of classroom composition, terrible curriculum and terrible instructional methods if the achievement (test scores) gap didn't break down along racial/ethnic lines.
well, I've certainly wondered that myself, but let me tell you: my town has very few black/Hispanic kids and in K-8 we are deeply committed to constructivism & homogeneous grouping.
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