Elementary school children who err on mathematics problems sometimes know more than they can say. These children explain their flawed math reasoning while using gestures that betray budding insights. They might just as well announce to their teachers: “Read my hands! I’m on the verge of some serious learning here!”
Over the past decade, psychologist Martha W. Alibali of the University of Wisconsin-Madison and her colleagues have tracked grade schoolers’ hand-to-mouth discrepancies in deciphering math problems.
“Gestures sometimes represent an emerging understanding of mathematical strategies that kids can’t yet articulate,” Alibali says. “Perhaps, we can provide instruction that more effectively brings out this leading edge of children’s knowledge.”
Alibali currently finds herself perched on the leading edge of a scientific movement that aims to redefine the study of children’s learning. Most members of this movement conduct microgenetic research, a term that refers not to a branch of the Human Genome Project but to the study of the origins of learning over periods ranging from minutes to weeks. Microgenetic work looks at how individuals move from one problem-solving strategy to another as they repeatedly confront a particular task during a series of trials.
Alibali’s team, for example, finds that the children who learn best tend to express a variety of problem-solving strategies using both words and gestures. When it comes to figuring out new math concepts, strategic variety is the spice of insight.
The microgenetic approach heralds a rebirth of research into children’s learning, says psychologist Robert S. Siegler of Carnegie Mellon University in Pittsburgh. For the past 30 years, scientists have mainly taken experimental snapshots of youngsters’ thinking capacities at different ages. For instance, studies have searched for the ages at which infants and toddlers first imitate others, use tools, and count small sets of objects. Much research now focuses on when children gain understanding of other people’s beliefs and intentions (SN: 9/16/95, p. 181).
This approach partly reflects the view–championed 50 years ago by French psychologist Jean Piaget–that children devote themselves to acquiring crucial stepping-stones of knowledge, such as realizing that quantities of items stay the same when they’re rearranged. Piaget characterized much of the rest of learning as the passive noting of associated items, such as the words adults use to refer to various objects.
Piaget’s ideas supported the influential theory that genetic programs control mental abilities, such as understanding another person’s beliefs, without the need for a lot of trial-and-error learning.
In contrast, microgenetic researchers regard everyday learning, which Piaget downplayed, as central to development. Four consistent findings have emerged from the approximately 50 microgenetic studies of learning conducted over the past decade, Siegler says.
First, learning usually occurs gradually, with tactical retreats as well as advances. After discovering more-effective strategies, children often resort to older ways of thinking about a task, at least for a while.
Second, children discover new thinking strategies while succeeding at a task, as well as while failing at it.
Third, children who come up with several problem-solving strategies, even wrong ones, frequently learn more than those who generate just one or two strategies, even correct ones.
Finally, children draw on intuitive knowledge about numbers and other matters as they learn.
Traditional developmental researchers want to narrow down children’s various attempts at solving specific problems. In experiments to explore learning, these scientists weed out such variability so that they can discern typical, age-specific thinking strategies. Such studies portray kids as moving, one step at a time, from simpler to more complex types of thought.
On closer inspection, this developmental staircase vanishes like a statistical mirage, Siegler contends. Microgenetic evidence shows that children usually make mental advances by riding “overlapping waves” of learning strategies, he says. At any one time, some strategies are cresting, some are waning, some are gaining renewed force, and new ones are forming just below the surface of conscious deliberations (SN: 1/2/99, p. 5).
“The coexistence of diverse strategies and ways of thinking in the same person is a central characteristic of our cognitive system,” Siegler contends. “This gives people the flexibility to adjust what they do to the demands of the problem and the situation.”
Grade school kids struggling with unfamiliar math problems provide a vivid look at how waves of thought–some conscious, some not–coalesce and diverge as they course toward insight, Alibali says.
Consider third and fourth graders working on math problems in which the two sides of an equation represent the same quantity. This concept is known as mathematical equivalence. Youngsters asked to fill in the blank in a problem such as 3 + 5 + 4 = ____ + 4 frequently answer incorrectly. Moreover, on initial trials, most of them describe faulty problem-solving strategies in both their words and gestures.
A child might say, “I added 3, 5, 4, and 4 and got 16 and put it in the blank” while successively pointing to 3, 5, and 4 on the left side of the equation, 4 on the right side, and then the blank. Without being told whether the answer’s correct, the child is given the next problem.
As more trials ensue, some children repeat their explanation of errors, while their gestures signal a different interpretation, Alibali holds. For example, a child who says, “I added 3, 5, 4, and 4 and got 16 and put it in the blank” now points only at 3 and 5, the two numbers that can be added to arrive at the correct solution.
Children who, for a brief period, express different strategies in gestures than in speech benefit more from instruction than those whose gestures consistently match their words, Alibali and her coworkers find. Moreover, kids who use three or more verbal problem-solving strategies in initial trials perform better after instruction and show a greater willingness to abandon earlier approaches than do those who begin with only one or two strategies.
Teaching techniques randomly assigned in these studies include asking a child who has incorrectly solved a problem to think of another way to solve it, briefly explaining the principle of mathematical equivalence, offering the analogy of two people on a teeter-totter to get a child to think about how to balance both sides of an equation, and telling a child that, since one number appears on both sides of the equation, the other two numbers can be added to get the correct answer.
Children instructed in these ways usually changed their strategies for the better in a halting fashion over repeated trials rather than in a sudden shift. Abrupt strategy changes occurred mainly among the children who initially used only one or two strategies and then learned the add-two-numbers procedure.
When given novel equivalence problems involving multiplication, the most accurate children were those who had exhibited several strategies in solving the addition problems and then gradually adopted effective tactics. The previous instruction in the add-two-numbers procedure had a smaller impact than the other teaching methods did. Abrupt learners in previous trials scored much lower on the multiplication problems.
“Gradual change may lead to more robust and flexible learning than abrupt change, particularly if the gradual change is accompanied by reflection about how old and new strategies relate to one another,” Alibali says. She and her colleagues have observed this among children who have a wide range of scores on intelligence tests.
In studies conducted with psychologist Susan Goldwin-Meadow of the University of Chicago, Alibali also finds that teachers are sensitive to students’ gestures and words. The researchers have videotaped five experienced teachers as they instruct individual fourth graders on mathematical equivalence.
When a youngster either displays matching gestures and words or doesn’t make any gestures, the teacher frequently repeats and rephrases what the child has said. In contrast, teachers seldom reiterate a child’s speech when the accompanying gestures don’t match. A student’s clashing gestures and words make it tougher for a teacher to interpret the child’s message, Alibali proposes.
Interestingly, children comprehend much more of what teachers say when the instructor uses matching gestures, as opposed to either mismatching gestures or no gestures at all. New or difficult concepts to children become clearer when teachers reinforce them in nonverbal ways, the researchers propose.
They describe their word-and-gesture research in an upcoming book, Microdevelopment: Transition Processes in Development and Learning (N. Granott and J. Parziale, eds., Cambridge University Press).
Other microgenetic findings suggest that infants and toddlers learn new mental strategies much as older children do. To determine whether an event violates a baby’s expectations, most current research times the baby’s gaze. In one such experiment, a child can see the researcher put two ducks and then one duck behind a screen. When the experimenter lifts the screen, babies look longer at the ducks if they see four of them than if they see only three. Some investigators view such results as evidence of basic counting abilities in infants.
It’s hard to know what, if anything, prolonged gazes imply about infants’ mental development, Siegler contends. Unfortunately, looking-time experiments create the impression that babies abruptly acquire knowledge about math and other topics, he says.
As with older children, microgenetic methods indicate that overlapping sets of problem-solving tactics in infants and toddlers foster gradual learning.
Siegler and Zhe Chen of the University of California, Davis described one such study last year in Across the Great Divide, a monograph of the Society for Research in Child Development. The researchers found that at both 1 1/2 and 2 1/2 years old, youngsters employ multiple strategies while learning to choose a tool that can pull a toy from the middle of a table.
For instance, the researchers gave a child access to a plastic rake, an oar-shaped toy, a striped rod, a toy cane with no handle, the cup end of a ladle, and the curved end of a cane. Only by gripping the end of the rake could the child both reach the center of the table and drag in a doll depicting Sesame Street’s Ernie.
Each of 42 younger and 44 older toddlers had five opportunities to use any of six tools to nab the toy. After their third try, youngsters were randomly assigned to watch an experimenter demonstrate how to use the correct tool to get the toy, receive hints from an experimenter who pointed to the correct tool (“Can you use this to get Ernie?”), or go on without instruction.
This process was repeated for a second set of tools and a new toy.
Finally, kids had three chances to snag a third toy with another tool array but no instructions.
Nearly all the children displayed three or four problem-solving strategies throughout the trials. These included asking their mother for help, playing with tools but not using them to reach for the toy, leaning forward and stretching an arm toward the toy, trying to bat the toy with an incorrect tool (which sometimes worked), and wielding the correct tool (which, from lack of practice, sometimes failed).
Even on their first few tries, most younger and older toddlers picked tools with heads and long shafts that might be useful for capturing a toy. Instruction, and especially demonstrations, boosted success for 1 1/2-year-olds. Still, they often resorted to their earlier tactics, such as pleading for Mom’s help.
Older toddlers profited even more from both types of instruction. On the third round of trials, those given either demonstrations or hints usually chose the correct tool and rarely tried other strategies.
Boys retrieved the toy more often than girls did. Siegler attributes this difference to the girls’, at least in this sample, initially having less interest in the toy tools. In a follow-up study, girls did as well as boys at getting the toy when all children were first encouraged by an experimenter to use the tools.
Another microgenetic study of infants has generated findings consistent with Siegler’s. As either crawlers or walkers, infants use a variety of strategies for going up and down ramps (SN: 3/20/99, p. 184). Crawlers with practice gradually learn to make better, safer choices. However, novice walkers revert to poor ramp-navigation strategies-such as heading straight down a steep slope-and must go through a new learning process.
Learners of all ages resort to multiple strategies to solve certain problems and take a halting, uneven path in adopting newly discovered tactics for those problems, says psychologist Deanna Kuhn of Columbia University. Kuhn and her colleagues have observed this pattern in grade-schoolers and college students given several weeks to learn to solve problems in social reasoning–such as inferring causes of other students’ academic failures from their school records–and scientific reasoning–such as determining which of a set of variables affects the speed of a toy boat being pulled through the water.
The consistency of microgenetic findings may spur efforts to improve classroom teaching, Siegler notes.
Educational psychologist Sharon Griffin of Clark University in Worcester, Mass., heads one such project for low-income students. She has developed a program that grade school teachers can use to instruct students in basic number concepts in preparation for learning math. The tasks include counting, realizing that the last number counted indicates the size of a set, and recognizing larger and smaller sets of items by their visual appearance.
Many middle- and upper-class kids enter school already equipped with this knowledge, largely because they’ve had regular number practice on a variety of childhood board games and in everyday conversations with adults about number-related topics, Griffin says.
Math achievement improved dramatically among 68 youngsters from low-income families after teachers used the number-sense program in kindergarten and first grade, Griffin says. By the end of first grade, these kids did as well on math tests as a group of wealthier peers receiving standard math instruction in another school. However, a group of inner-city kids who didn’t receive number-sense instruction but had regular math instruction scored progressively worse on math tests in the first 2 years of school, Griffin says.
Teachers can sometimes heighten their impact by simply inviting students to think in new ways. Laboratory-based microgenetic studies indicate that in a variety of academic subjects, asking children to explain both why correct answers work out and why incorrect answers don’t produces much more learning than focusing only on correct answers does, Siegler says.
Children who hash out both their correct and incorrect answers tend to generate a greater number of new problem-solving strategies, from which they can select those with the broadest applications, he contends.
Questions remain about the extent to which microgenetic studies tap into how kids learn in real-world settings. Also, some childhood mental feats may not spring from a wide enough spectrum of thinking strategies to justify microgenetic analyses.
Still, there’s much promise in the microgenetic emphasis on learning as integral to mental development, says psychologist Rachel K. Clifton of the University of Massachusetts in Amherst. “Microgenetic research is hard to do, but it’s valuable,” she remarks. “It requires a shift of perspective from many developmental researchers.”
Siegler hopes that his unconvinced colleagues will change their perspective soon, although he knows by now to look for halting, gradual transformations.