Why kids should use their fingers in Math class

Evidence from brain science suggests that far from being “babyish,” the technique is essential for mathematical achievement.

A few weeks ago I (Jo Boaler) was working in my Stanford office when the silence of the room was interrupted by a phone call. A mother called me to report that her 5-year-old daughter had come home from school crying because her teacher had not allowed her to count on her fingers. This is not an isolated event—schools across the country regularly ban finger use in classrooms or communicate to students that they are babyish. This is despite a compelling and rather surprising branch of neuroscience that shows the importance of an area of our brain that “sees” fingers, well beyond the time and age that people use their fingers to count.

In a study published last year, the researchers Ilaria Berteletti and James R. Booth analyzed a specific region of our brain that is dedicated to the perception and representation of fingers known as the somatosensory finger area. 

Philippe Lissac / Godong / Corbis

Remarkably, brain researchers know that we “see” a representation of our fingers in our brains, even when we do not use fingers in a calculation. The researchers found that when 8-to-13-year-olds were given complex subtraction problems, the somatosensory finger area lit up, even though the students did not use their fingers. This finger-representation area was, according to their study, also engaged to a greater extent with more complex problems that involved higher numbers and more manipulation. Other researchers have found that the better students’ knowledge of their fingers was in the first grade, the higher they scored on number comparison and estimation in the second grade. Even university students’ finger perception predicted their calculation scores. (Researchers assess whether children have a good awareness of their fingers by touching the finger of a student—without the student seeing which finger is touched—and asking them to identify which finger it is.)

Evidence from both behavioral and neuroscience studies shows that when people receive training on ways to perceive and represent their own fingers, they get better at doing so, which leads to higher mathematics achievement. The tasks we have developed for use in schools and homes (see below) are based on the training programs researchers use to improve finger-perception quality. Researchers found that when 6-year-olds improved the quality of their finger representation, they improved in arithmetic knowledge, particularly skills such as counting and number ordering. In fact, the quality of the 6-year-old’s finger representation was a better predictor of future performance on math tests than their scores on tests of cognitive processing.

Many teachers have been led to believe that finger use is useless and something to be abandoned as quickly as possible.

Neuroscientists often debate why finger knowledge predicts math achievement, but they clearly agree on one thing: That knowledge is critical. As Brian Butterworth, a leading researcher in this area, has written, if students aren’t learning about numbers through thinking about their fingers, numbers “will never have a normal representation in the brain.”

One of the recommendations of the neuroscientists conducting these important studies is that schools focus on finger discrimination—not only on number counting via their fingers but also on helping students distinguish between those fingers. Still, schools typically pay little if any attention to finger discrimination, and to our knowledge, no published curriculum encourages this kind of mathematical work. Instead, thanks largely to school districts and the media, many teachers have been led to believe that finger use is useless and something to be abandoned as quickly as possible. Kumon, for example, an after-school tutoring program used by thousands of families in dozens of countries, tells parents that finger-counting is a “no no” and that those who see their children doing so should report them to the instructor.

Stopping students from using their fingers when they count could, according to the new brain research, be akin to halting their mathematical development. Fingers are probably one of our most useful visual aids, and the finger area of our brain is used well into adulthood. The need for and importance of finger perception could even be the reason that pianists, and other musicians, often display higher mathematical understanding than people who don’t learn a musical instrument.

Teachers should celebrate and encourage finger use among younger learners and enable learners of any age to strengthen this brain capacity through finger counting and use. They can do so by engaging students in a range of classroom and home activities, such as:

Give the students colored dots on their fingers and ask them to touch the corresponding piano keys:

youcubed.org

youcubed.org

Give the students colored dots on their fingers and ask them to follow the lines on increasingly difficult mazes:

(The full set of activities is given here.)

The finger research is part of a larger group of studies on cognition and the brain showing the importance of visual engagement with math. Our brains are made up of “distributed networks,” and when we handle knowledge, different areas of the brain communicate with each other. When we work on math, in particular, brain activity is distributed among many different networks, which include areas within the ventral and dorsal pathways, both of which are visual. Neuroimaging has shown that even when people work on a number calculation, such as 12 x 25, with symbolic digits (12 and 25) our mathematical thinking is grounded in visual processing.

A striking example of the importance of visual mathematics comes from a study showing that after four 15-minute sessions of playing a game with a number line, differences in knowledge between students from low-income backgrounds and those from middle-income backgrounds were eliminated.

Number-line representation of number quantity has been shown to be particularly important for the development of numerical knowledge, and students’ learning of number lines is believed to be a precursor of children’s academic success.

Visual math is powerful for all learners. A few years ago Howard Gardner proposed a theory of multiple intelligences, suggesting that people have different approaches to learning, such as those that are visual, kinesthetic, or logical. This idea helpfully expanded people’s thinking about intelligence and competence, but was often used in unfortunate ways in schools, leading to the labeling of students as particular type of learners who were then taught in different ways. But people who are not strong visual thinkers probably need visual thinking more than anyone. Everyone uses visual pathways when we work on math. The problem is it has been presented, for decades, as a subject of numbers and symbols, ignoring the potential of visual math for transforming students’ math experiences and developing important brain pathways.

It is hardly surprising that students so often feel that math is inaccessible and uninteresting when they are plunged into a world of abstraction and numbers in classrooms. Students are made to memorize math facts, and plough through worksheets of numbers, with few visual or creative representations of math, often because of policy directives and faulty curriculum guides. The Common Core standards for kindergarten through eighth grade pay more attention to visual work than many previous sets of learning benchmarks, but their high-school content commits teachers to numerical and abstract thinking. And where the Common Core does encourage visual work, it’s usually encouraged as a prelude to the development of abstract ideas rather than a tool for seeing and extending mathematical ideas and strengthening important brain networks.

To engage students in productive visual thinking, they should be asked, at regular intervals, how they see mathematical ideas, and to draw what they see. They can be given activities with visual questions and they can be asked to provide visual solutions to questions. When the youcubed team (a center at Stanford) created a free set of visual and open mathematics lessons for grades three through nine last summer, which invited students to appreciate the beauty in mathematics, they were downloaded 250,000 times by teachers and used in every state across the U.S. Ninety-eight percent of teachers said they would like more of the activities, and 89 percent of students reported that the visual activities enhanced their learning of mathematics. Meanwhile, 94 percent of students said they had learned to “keep going even when work is hard and I make mistakes.” Such activities not only offer deep engagement, new understandings, and visual-brain activity, but they show students that mathematics can be an open and beautiful subject, rather than a fixed, closed, and impenetrable subject.

Some scholars note that it will be those who have developed visual thinking who will be “at the top of the class” in the world’s new high-tech workplace that increasingly draws upon visualization technologies and techniques, in business, technology, art, and science. Work on mathematics draws from different areas of the brain and students need to be strong with visuals, numbers, symbols and words—but schools are not encouraging this broad development in mathematics now. This is not because of a lack of research knowledge on the best ways to teach and learn mathematics, it is because that knowledge has not been communicated in accessible forms to teachers. Research on the brain is often among the most impenetrable for a lay audience but the knowledge that is being produced by neuroscientists, if communicated well, may be the spark that finally ignites productive change in mathematics classrooms and homes across the country.

(Source: The Atlantic)

The hundred trillion stories in your head

For the father of modern neuroscience, cellular anatomy was like the most exciting fiction, writes Benjamin Ehrlich, the author of The Dreams of Santiago Ramón Cajal, in the Paris Review. Read on: 

Fiction is, by definition, a world away from fact—but Santiago Ramón y Cajal, often heralded as “the father of modern neuroscience,” used it to find objective truth. Cajal spent his days at the microscope, gazing down at faint, entangled fibers that appeared to his fellow anatomists as inscrutable labyrinths. Contrary to prevailing theory, the Spaniard discerned that the nervous system, including the brain, comprises distinctly individual cells (neurons), which, he theorized, must communicate across the infinitesimal spaces between them (synapses).

It was Cajal who first applied the term plasticity to the brain; he went so far as to recommend “cerebral gymnastics” for mental enhancement, presaging twenty-first century insights and trends about brain exercise. “If he is so determined,” Cajal said, “every man can be the sculptor of his own brain.” If all Russian literature comes from Gogol’s “Overcoat,” and all modern American literature comes from “a book by Mark Twain called Huckleberry Finn,” then international brain research, including grand projects like the BRAIN Initiative and the Human Brain Project, emerges from the unlikely work of Santiago Ramón y Cajal.

Santiago Ramon Y Cajal, the father of modern neuroscience.

Cajal was born in 1852, high in the mountains of northern Spain; his head always seemed to belong in the clouds. The landscape of his childhood was epic. Aragonese folklore echoed through dust-colored pueblos, swept through by the specters of conquests and kings. The young Cajal idolized these legendary figures, maybe because village life was incessantly prosaic.

Alto Aragón was notoriously inhospitable; the highland region that Robert Hughes, in his biography of Goya (another native son), noted for its “sour wine, straw bedding, tough meat”—“semi-troglodytic conditions.” Almost nothing grew from the callous, fissured soil; Cajal’s home was a ramshackle pile of cobblestone. “Not a flower pot in the windows,” he recalls in his autobiography, Recollections of My Life: “not the smallest decoration on the fronts of the houses, nothing in a word, to indicate the slightest feeling for beauty.”

Cajal’s father was a man of facts, implacable and austere. The son of peasant farmers, he’d abandoned his home at age twelve, in search of less desolate fields. He apprenticed to a lowly barber-surgeon and went on to achieve a medical degree, the triumph of a grueling life. There was no time for leisure, for distractions from the path. The human mind, Cajal’s father believed, was made for obtaining knowledge. “He repudiated or despised all culture of a literary or of a purely ornamental or recreative nature,” Cajal recalls. Only medical books were allowed in the house—absolutely no fiction. Art, Cajal’s father believed, was the symptom of a devastating illness.

But Cajal’s mother, a covert romantic, kept cheap fantasy novels hidden at the bottom of a trunk and smuggled them into the hands of her children, to their delight. As the local surgeon, Cajal’s father was often away from home, and when he returned, his wife’s “excessive softness” enraged him; he was chagrined to recognize it in his son, too. He tried to eradicate the boy’s literary impulses, punishing him with whips, cudgels, and other instruments of torture often seen in “penny dreadful” horror tales and haunting children’s nightmares.

The abuse only seemed to embolden Cajal’s creative instincts. “Whenever I had finished supper,” he recalls, “I eagerly hastened to my little room and, until I fell asleep, spent my time giving form and life to the jumble of stains on the wall and the cobwebs of the ceiling, which I transformed, by the power of thought, into the wings of a magic stage, across which filed the cavalcade of my fantasies.” His mind transmuted shadows into story: a perceptual trick that would later help clarify his view of even more profound obscurities.

Cajal convinced his father that he needed a quiet room of his own in which to study. The pigeon house next to the family barn became the site of his first experiments, where he was free to imagine and explore. A window opened onto the roof of a neighbor, a pastry chef with slightly more elevated tastes. In the attic, among the sweetmeats and dried fruit, Cajal espied a smorgasbord of literary treats. He lost himself in Dumas, Hugo, Cervantes, and others. Long before peering through microscopes, Cajal trained his eyes to the pages of books. Reading novels primed his mind to explore more invisible realms.

After high school, Cajal’s father enrolled him in medical school, where the only subject that held his interest was anatomy. Though it had emerged decades earlier, cell theory was revolutionizing—or scandalizing—the field then. Reading about it, Cajal encountered literary metaphors that drew him in, such as the famous line from the German pathologist Rudolf Virchow: “The body is a state in which every cell is a citizen.” Cajal’s first look through a microscope confirmed this idea, showing him, in his own words, “captivating scenes from life of the infinitely small.” For twenty continuous hours—or so he claimed—he watched the movement of a leucocyte away from a capillary, akin, in his vivid imagination, to high-stakes escape. He even wrote and illustrated a novel about a miniature man—about the size of a cell—traveling through bodies of gargantuan beings on Jupiter.

Cellular anatomy, for Cajal, was like the most exciting fiction. He became a researcher, setting up a makeshift laboratory—not unlike the old pigeon house—in his parents’ attic. Later, he witnessed a demonstration of the so-called black reaction, an erratic chemical stain—which other biologists had abandoned—that had the power to reveal whole hidden elements beneath the discriminating light of the microscope. Cajal was stunned by the clarity of the results; the arcane forms reminded him of ink on paper. He immediately applied the technique to the nervous system—the holy grail of human anatomy—hoping to uncover “the material course of thought and will.”

Cajal modeled his scientific quest after literary heroes from his childhood. He cast himself in the mold of Crusoe, envisioning the brain as “a world consisting of a number of unexplored continents and vast stretches of unknown territory,” and devoting his work to “islands of discovery.” In 1888, working alone at the microscope in his home laboratory, Cajal observed the endpoint of a nerve fiber, a nearly imperceptible phenomenon that led him to declare that nerve cells were independent—thus beginning the saga of modern neuroscience. “The job of the anatomist,” Cajal writes, “is to separate the apparent from the real.” Like his beloved Don Quixote, he believed in an alternative view of the world; unlike Quixote, he succeeded in rallying the people around him, who eventually celebrated his foresight.

How did Cajal see what others could not? To anthropomorphize was his genius—yet such an approach seemed unscientific, to say the least. “Cajal treated the microscopic scene as though it were alive,” recalled Charles Sherrington, his lifelong friend and fellow Nobel Laureate, “and were inhabited by beings which felt and did and hoped and tried even as we do.” Sherrington saw that fantasy aided the unorthodox Spaniard’s ability to visualize the brain. “If we would enter adequately into Cajal’s thought in this field,” Sherrington continues, “we must suppose his entrance, through the microscope, into a world populated by tiny beings actuated by motives and striving and satisfactions not very remotely different from our own.” Cajal highlighted that subject and object—the brain scientist and the neuron—descended from the same evolutionary ancestor, contained the same physical material, and were beholden to the same mortal laws.

The world of the infinitely small, like novels of his youth, seemed more real to Cajal than everyday existence. How else could he routinely spend up to fifteen hours a day, for almost fifty years, alone in the laboratory? He claimed to have observed a million neurons, witnessing them in every phase of their lives: birth, development, movement, relationships, adversity, trauma, decline, and death. On thumbnail sheets of dead brain tissue, the Spaniard’s cherished stories came to life. He imagined neurons as protagonists in an intense cerebral drama. Their fibers “groped to find another.” Their aching contacts became “protoplasmic kisses”—“the final ecstasy of an epic love story.”

More than a century after his 1906 Nobel Prize, Cajal’s portraits are iconic. It might be natural to assume that these drawings are copies of what he saw through the eyepiece, like cerebral still lifes. Yet his illustrations were strikingly interpretative and deeply personal, blurring the line between abstraction and representation. After viewing many cells, Cajal distilled their features to a type. His images are composites, highlights, even exaggerations of actual neurons; I’ve heard it said that he often drew from memory after taking long walks through the park. Cajal called them “pieces of reality,” these tactfully fictionalized depictions. Had he listened to his father, forsaking literature, he might never have recognized the truth. The human brain contains around eighty billion neurons, we now know, each of which may interact with up to ten thousand others. One might wonder: Do we tell their ephemeral stories, or do they, somehow, tell ours?