kitchen table math, the sequel: the shuffling of mathematics problems improves learning

Saturday, September 1, 2007

the shuffling of mathematics problems improves learning

Haven't read yet, but here's the abstract.

Abstract In most mathematics textbooks, each set of practice problems is comprised almost entirely of problems corresponding to the immediately previous lesson. By contrast, in a small number of textbooks, the practice problems are systematically shuffled so that each practice set includes a variety of problems drawn from many previous lessons. The standard and shuffled formats differ in two critical ways, and each was the focus of an experiment reported here. In Experiment 1, college students learned to solve one kind of problem, and subsequent practice problems were either massed in a single session (as in the standard format) or spaced across multiple sessions (as in the shuffled format). When tested 1 week later, performance was much greater after spaced practice. In Experiment 2, students first learned to solve multiple types of problems, and practice problems were either blocked by type (as in the standard format) or randomly mixed (as in the shuffled format). When tested 1 week later, performance was vastly superior after mixed practice. Thus, the results of both experiments favored the shuffled format over the standard format.

Instructional Science ($)
published online April 19, 2007

the Saxon shuffle
[A] very small number of mathematics textbooks use what we call a shuffled format (e.g., Saxon, 1997). A textbook with a shuffled format may have lessons identical to those in the standard format, and moreover, the two formats need not differ in either the number of practice sets within the text or the number of practice problems per practice set. But, with the shuffled format, the practice problems are systematically arranged so that practice problems are both distributed and mixed. For example, after a lesson on the quadratic formula, the immediately following practice set would include no more than a few quadratic formula problems, with other quadratic formula problems appearing in subsequent practice sets with decreasing frequency. Thus, the practice problems of a given type are systematically spaced throughout the textbook. This spacing intrinsically ensures that the problems within each practice set include a mixture of different types, as there are no more than one or two practice problems of each kind within each practice set. In order to achieve such variety in the early portion of the textbook, the first several practice sets can include problems relating to topics covered in previous years.


Perhaps the most well known example of the shuffled format is the Saxon line of mathematics textbooks (e.g., Saxon, 1997). In these textbooks, no more than two or three problems within each practice set are drawn from the immediately preceding lesson, and the remaining one or two dozen problems are drawn from many different lessons. We are not aware of any published, controlled experiments comparing a Saxon and non-Saxon textbook, but such an experiment may not be particularly informative because it would be confounded by the numerous differences between any two such texts. That is, regardless of the outcome of an experimental comparison of a shuffled textbook and a standard textbook, any observed differences in, say, final exam performance might reflect differences in the lessons rather than practice format.

Such confounds would be avoided, however, if two groups of students were presented with the same lessons and different practice sets. For example, each group of students could receive a packet that includes the lessons from a traditional textbook, and these lessons would appear in the same order for both groups. Both groups would also see the same practice problems, but the problems would be arranged in either a standard format or shuffled format. By way of disclosure, neither author has an affiliation with a publishing company or mathematics textbook, although the first author is a former mathematics teacher who has taught with textbooks from many different publishers, including Saxon.

There is no doubt in my mind -- none -- that shuffled problems produce better retention than massed problems.

overlearning overrated?
how long does learning last?
shuffling math problems is good
Saxon rules
Ken's interval
same time, next year
remembering foreign language vocabulary


KDeRosa said...

I would think it is even more efficient to systematically fade both the amount of problem and the interval between problem sets for each new problem type.

For example after a new problem type is learned via massed practice, subsequent practice sets might go something like this:

day 2: problems 10
day 3: problems 10
day 4: problems 8
day 5: problems 8
day 6: problems 6
day 7: problems 6
day 9: problems 6
day 11: problems 6
day 13: problems 6
day 16: problems 5
day 20: problems 5
day 25: problems 5
day 30: problems 5
day 40: problems 4
day 50: problems 4
day 60: problems 4
day 80: problems 3
day 100: problems 3

That's how the the spiral should work. As the student better learns the material, it gets refreshed at increasingly larger before the knowledge has a chance to fade from memory.

This is how Engelmann does it in all the DI programs. The only exception is when a subskill gets subsumed into a more complicated skill (once it has been mastered), then only the more complicated skill gets the distributed practice.

Catherine Johnson said...


I'll send you the two articles --