How The Mind Deals with Math (Special Presentation)
By Dr. Eugene Maier
Most of what I say here comes from three books. The major source is The Number Sense (1997). The author, Stanislas Dehaene, is a neuropsychologist with a background in mathematics. He was a researcher at the Institute of Health and Medical Research in Paris at the time this book was written. A Celebration of Neurons, An Educators Guide to the Human Brain (1995) was written by Robert Sylwester. Bob, a former colleague at the University of Oregon, writes and lectures on what's known about the brain and the implications for educators. The Psychology of Invention in the Mathematical Field (1944), by Jacques Hadamard, is the third source. Hadamard, a French mathematician, is best known for his part in proving the so-called prime number theorem. He fled France for the United States during the Nazi occupation, returning in 1944. His book is the outgrowth of a lengthy questionnaire sent to mathematicians asking them to describe their thinking. Originally published by Princeton University Press, his book is now available as a Dover reprint. Hadamard died in 1963 at the age of 97.
Everybody Counts, and Adds, and Subtracts
Show a five-month old baby slides of two objects and measure how long they look at it (Dehaene). The time remains constant or diminishes as habituation sets in. Suddenly change the number of objects to three. The time the baby watches shows a marked increase. The increase depends on the number of objects, not on the type or location of the objects. The increase in attention appears to be dependent only on numerosity, sometimes referred to as cardinality. In a similar vein, show a baby an object behind a screen, and then another object behind a second screen. Remove the screen and measure the time the baby fixates on the collection of objects. The time increases if an object has either been taken away or added behind the screen before the screens are removed. The baby fixates longer if there are one or three objects shown rather than the expected two. Similarly, a baby is shown two objects, a screen is placed over one, and then the other is removed in sight of the baby. If the screen is then removed, the baby will fixate much longer if the removed screen reveals two objects rather than one. Dehaene concludes from these experiments and others that babies are born with an "innate, abstract competence for numbers." There is in the human being (and also in animals) an innate intuition for number. This exhibits itself in babies by their ability to distinguish numerosity between sets of objects and rudimentary knowledge of addition and subtraction.
As language develops, in Dehaene's words, children also show "precocious competence" in counting. By about three-and-one-half years, children know that in the counting process the names of numerals occur in a particular order, and that in counting a set of objects the order in which you point to them doesn't matter as long as each object is touched exactly once. At first the child apparently doesn't know that counting provides the answer to the question "how many?" If you watch a three-year old count the number of Easter eggs they have collected and you ask them, "How many eggs do you have?" they won't necessarily give you the number they reached. However, by four years of age they are aware that counting provides the answer to the question "How many?" Children also develop their own means of adding and subtracting, mostly by counting on their fingers. They find shortcuts, like commutativity, and hone their strategies, taking into account time and reliability of the process. All this arithmetical activity develops without explicit instruction, apparently based on intuitive understanding of number and the meaning of number calculations.
There's No Black Box
How does our mind operate? What is going on in our brain that manifests itself in this intuitive knowledge of number? Brain imaging techniques reveal that many areas of the brain are involved in arithmetical tasks. To quote Dehaene, "arithmetic is not…associated with a single calculation center. Each operation recruits an extended cerebral network. Unlike a computer, the brain does not have a specialized arithmetic processor…Even an act as simple as multiplying two digits requires the collaboration of millions of neurons distributed in many brain areas" (p 221). Dehaene concludes that in addition to not having a central processing unit—a black box that is a calculation center—there is another significant difference between the neural architecture of our brain and the modern computer. He maintains that the brain is not a digital device but is analogic in the manner in which it perceives quantities—like an analog rather than a digital computer—more like an hour glass than a digital watch. Furthermore, our perception is "fuzzy." We are not bad at approximation and judging differences that are large but our perception is not precise. Studies show that human beings can recognize one, two or three dots at a glance. The time required to identify the number of dots grows rapidly after two or three. We can improve on our approximations with practice, but no amount of practice will enable us to say at a glance, with accuracy, "there are 105 dots." For a digital device, finding the exact number would be as easy as finding an approximation. The more distant two numbers, the quicker we are at determining which of two is larger. For example, we are quicker at determining that 982 is larger than 126 than we are at determining that 272 is larger than 267. To a digital computer the difference between two numbers doesn't matter. Another factor that distinguishes our brain from a computer is that we are always making associations and analogies. If I gave you the task of approximating the number of dots on a page, you couldn't help but notice that the dots were arranged in the shape of a banana. A computer scanning the banana shape counts 107 dots without awareness of the banana shape.
How are You Feeling
According to Dehaene and Sylwester, emotions and reason are tightly linked in our cerebral structure and emotions often get the upper hand. That's not a bad thing. When danger was felt, strong survival instincts led our ancestors, and leads us, to flee without cogitating. Better, as Bob Sylwester points out, "to flee unnecessarily many times than to delay once for a more detailed analysis of the threat and so die well informed." But it does lead to impulsive, and as we often say, irrational behavior. You can probably cite your own examples. I remember smelling smoke at the dinner table one evening. Looking for the source, our oldest child went downstairs to check his bedroom. He came running upstairs to announce the basement was full of smoke. I took one peek downstairs, called the fire department and hurried everyone out of the house. Only when we were standing in the driveway waiting for the fire truck did someone remember that Jon was still in the dining room, strapped in his high chair. It is well accepted that emotional response can impede rational thought. Most of us who teach math are well aware of the effects of mathephobia. Dehaene is convinced that "children of equal initial abilities may become hopeless or excellent at math depending on their love or hatred of the subject." He maintains, "Passion breeds talent." Sylwester suggests one reason emotion is such a powerful force in our behavior is that far more neural fibers project from the limbic system where our emotions are centered, than into the cortex where logical, rational thought is centered. One might say the emotions have more impact on rational thought than rational thought has on emotions. Apparently sensory perception also makes its way to the emotional centers more rapidly than to the rational thought centers. Our emotions get a head start in our reactions to the external world.
We are Sense-ible, and Reflective, and Creative
We perceive the external world through our senses. Receptors that receive and convert stimuli into neural codes are dense throughout our sense organs—250 in a patch of skin the size of a quarter. The eyes predominate, containing some 70 percent of our body's receptors. About 30 percent of our brain is devoted to visual information. We perceive from the external world around us and we reflect on the input. Not only do we reflect, but we create—we string words together to make sentences. We combine sentences into lectures, perhaps expressing thoughts we have never had before. Where do they come from? Where does the sudden insight come from? The forgotten name that comes to us out of the blue; the solution to the problem we were working on yesterday that hits us while we are taking our morning shower? Dehaene begs the question with the comment that the "flash of invention is so brief that it can hardly be studied scientifically." Not only is it brief but it doesn't happen on demand. One can attach a bunch of electrodes to my neuronal fields and ask me to compare numbers, solve an equation, or estimate the number of dots on a screen. I can do that and the experimenter can measure what is happening in my brain. However, he can't ask me to have an "aha" or to remember a name I have forgotten or to suddenly see the solution of a problem I have been working on. Someday we may be able to—Dehaene holds the hope that there are physiological traces of neuronal activity below the "threshold of consciousness" that can be measured with brain imaging tools. Even though for the present we are unable to measure and locate the brain activity that occurs during the process of insight and invention, we can reflect on our thinking and the circumstances of our own creative thought. This is what Hadamard fifty years ago asked his colleagues to do. He sent them a long questionnaire about their habits and work style, about the circumstances leading up to and surrounding moments of insights. In the last question, and almost as an afterthought, he asked them to describe their mechanism of thought. Sifting over the replies and drawing on his own experiences, Hadamard identified four stages in the process of mathematical invention. He called these preparation, incubation, illumination and verification. Briefly, preparation is conscious thinking about a problem. Incubation is letting the problem sit without conscious thought. Illumination is the moment when the lights go on, and verification is putting together the rational justification for the insight.Hadamard based his conclusions on experiences such as the following, reported by Henri Poincaré. Poincaré had been searching for a set of functions that satisfied certain conditions. He continued his search for a fortnight when he interrupted his endeavors to go on an excursion with a group of people. He reported:
The incidents of the travel made me forget my mathematical work…we entered an omnibus to go some place or other. At the moment I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define…were identical with those of non-Euclidian geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return…I verified the result at my leisure.
Then I turned my attention to the study of some arithmetical questions without much success…Disgusted with my failure, I went to spend a few days at the seaside and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness and immediate certainty,…(pp. 13-14)
On the matters of thought mechanisms, Hadamard decided that most mathematicians are visual thinkers and their thought process entailed images other than mathematical symbols. The one notable exception he mentions is George Birkhoff who said he was "accustomed to visualizing algebraic symbols and to work with them mentally."The most celebrated response Hadamard received was from Albert Einstein. Einstein's reply is printed in its entirety in the appendix of Hadamard's book. He describes his thought mechanism this way:
The words or the language as they are written or spoken, do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced and combined…The above mentioned elements are, in my case, of visual and some of muscular type. Conventional words or signs have to be sought for laboriously only in a secondary stage. (pp. 142-243)
The only reference Dehaene cites in his brief discussion of the flash of mathematical invention is Hadamard's book. Apparently not much new has been added in the last fifty years.
Human beings are capable of storing vast amounts of information. We have particularly strong visual memories. We can spot a friendly face in a crowd, distinguishing it from hundreds of other faces, even if we haven't seen our friend for ten years. We are not so good at other kinds of memory. What would we do without name badges at conferences? We have all had the experience of seeing a familiar face and not being able to recall how or where we met. Emotion and memory are closely connected. The limbic system, the part of our brain that processes emotion, also plays an important role in processing memory. We recall emotions along with events and a particular emotion can evoke the memory of an event surrounding that emotion. Events that evoke strong emotion evoke strong memories. I remember a near disastrous event, and the accompanying feelings of terror, as if it occurred yesterday, yet it happened over 35 years ago. Then there are things we once knew and have forgotten. I suspect all of us can recall mathematical procedures that fall into this category. Tests have shown, however, that it is quicker to refresh memory than learn something for the first time. Associations also bring to mind past memories. Reunions are good at that, and the conversations may go, "Remember when so and so did such?" and "I haven't thought of that for years!" and on and on it goes. Dehaene says our memory is associative. That has its good points and its bad points, depending on the task at hand. Associative memory enables us to put together strings of recollections, but it also leads us to focus on bananas instead of dots, and it gets in the way of memorizing arithmetical facts. Experiments have shown that children regress in the time it takes them to recall addition facts once they begin to learn multiplication facts. The theory is that one begins to associate 2+3 with
2 x 3; while 8 and 9 are closely associated in our mental picture of the number line, we only confuse ourselves if we do that when we are taking a timed test on multiplication facts. Dehaene offers a lengthy discussion about the difficulty in remembering a sheet of multiplication facts. It's not that we can't do it; if all else fails, drill it into verbatim memory, much as one does a nonsense rhyme. What is sacrificed is meaning. I know things verbatim that I can rattle off without a bit of awareness of the meaning of the words that I am saying. An example is The Pledge of Allegiance that we recited every morning at the elementary school I attended.Mathematics class can become a matter of rote learning in which one memorizes how to carry out procedures without any sense of what is going on, yet with sufficient skill to pass the course. Carl Jung got good grades in algebra by mimicking what the instructor was doing but understanding nothing. He said he swindled his way through math.
Implications for Teaching and Learning
What are the implications the above observations have for teaching and learning? We can begin by fitting educational practices to how our mind works as opposed to trying to fit how our mind works to educational practices:
Children come to us with innate, intuitive knowledge of mathematics. We should build on that strength and be educators in the true meaning of the word. We should educe, that is draw out, and nurture this knowledge. And, by all means, avoid violating it. This means that we make fewer commands, "This is how you do it." We issue more invitations, "How would you do this?" Violating a child's intuition can happen in very subtle and unintentional ways. I once visited a first grade class during arithmetic period. The children were learning how to write the numerals. The teacher had placed a collection of dots on the board and asked for a volunteer to connect the dots to form the figure 5. A little girl volunteered; she started in the upper right corner and connected the dots in a continuous motion without lifting the chalk. What she drew looked like a 5 but the teacher, as nicely as she could, informed the class this wasn't the correct way to write a 5, and, drawing a second set of dots on the board, asked if someone else would like to try. A second child came forward, and starting in the upper left, drew the vertical portion and the bottom curve of the figure. Lifting the chalk, she then drew the top line from left to right. Yes, said the teacher and recited a little poem that described the "proper" way for making a 5, ending by drawing the "cap" on top. "My," I thought, "this is how school math becomes a mysterious and arcane subject." A first-grader does something that makes sense to them, and gives the proper result. Yet without explanation, they are told they're wrong! What impression can this make? Only that school math is an odd subject that has its own set of arbitrary, non-intuitive rules. Remember that children's number intuition is way ahead of their language skills. This means children will have intuitive knowledge of how to do something and not be able to explain what it is they did. A number of years ago as part of a project to bring college professors into elementary schools, I spent an hour a day teaching math to fifth-graders. Rusty was an alert, quiet child who sat in the back of the class, but he was on top of everything I did. He invariably arrived at correct solutions to problems I posed, often only recording an answer and perhaps a few isolated calculations. When I asked him how he arrived at an answer his usual response was something like, "I just knew." I, having acquired the common belief that an answer isn't acceptable unless one can explain how one arrived at it, kept pushing Rusty for explanations, which only frustrated both of us. Finally, it became clear to me that knowing something and explaining it are two different things. I quit asking Rusty how he arrived at his answers. I didn't want him to think his answers were unacceptable or incorrect because he couldn't explain how he arrived at them. Above all, I didn't want to undermine the marvelous intuitive understanding Rusty had of numbers and how they worked. So, rather than ask Rusty to explain his thinking, if I suspected Rusty had a misunderstanding, I would pose a similar question or two, changing the parameters slightly. If Rusty dealt with those correctly, I felt confident he knew what he was doing. Ever since I met Rusty, I've attempted to never give the impression that a student's work is incorrect or unacceptable because they can't explain how they arrived at their conclusion. As a matter of fact, I have found that sometimes when a student has taken a novel approach to a solution it is I, not the student, who doesn't understand.
A large part of our brain is devoted to sensory input and its processing—and everything seems to be connected to everything else. Doesn't it make sense to get as much sensory input as we can into our math instruction? This means handling and exploring things. There are lots of computer simulations available. For example, you can find programs that emulate geoboards or base ten pieces. But my sense is that they are not as good as handling the real thing. Before using one of those programs, I would have students physically move pieces and arrange rubber bands. I would always involve as many senses as possible. Maybe we ought to have scratch and sniff base ten pieces so students also use their sense of smell!
When mathematics is learned by rote, meaning is lost, and conversely, when meaning is absent, mathematics is learned by rote. If that is a concern, math should be taught in a context, a frame of reference that is meaningful to the learner. Meaning may not be a concern if the only goal is passing tests! Dehaene says the child's brain is not a sponge, it is a "structured organ that acquires facts only insofar as they can be integrated into its previous knowledge." Dehaene talks at length about the difficulty of remembering the times table. One problem is that there is no meaning attached to them. Adults, much less children, have a hard time telling you the meaning of the phrase "eight times seven is fifty-six." Few realize that fifty-six means "5 tens and 6 ones," so the multiplication fact "8 x 7 = 56" is simply a report that if seven 8's are arranged in groups of tens, one gets 5 tens with 6 left over. Once children understand about grouping tens, they can construct their own multiplication tables. (The English language doesn't help. Children in China have much less difficulty with grouping concepts. If counting in English were similar to counting in Chinese, we would, for example, read 56 as "five tens and 6" and 16 as "one ten and 6.") Some think that one creates a frame of reference for a mathematics topic by connecting to some part of the world outside of school. Because something comes from the world outside of school does not mean it creates a frame of reference that is meaningful to the student. For example, knowing how to compute board feet may be crucial if you work in a lumberyard. But, if I walk into the classroom and tell that to my students, give them a formula for computing board feet, and then ask them to solve related problems, I suspect that for most of them I have only promoted rote learning. Furthermore, if I took them to the lumberyard where I worked earning money for college, and expected them to compute board feet in their head while loading an order, I think most of them would be lost. If you want to learn about board feet, go to a lumberyard or bring the lumberyard to the classroom. Dealing with board feet can be instructive since one encounters lots of fractions, but it is difficult to bring the working lumberyard to the classroom or vice versa. To provide a context for studying a mathematical topic like fractions, it isn't necessary to relate it to an application from the world outside of school. One can create a context for fractions by using egg cartons or manipulatives created for the purposes of studying fractions, such as fraction bars or segment strips. The context provides a frame of reference in which students can become familiar with fractions and devise ways of dealing with them, while developing intuitive understanding. When mathematics becomes disconnected from students' intuitive understanding, the result is innumeracy. We are by nature numerate; numeracy is built into human beings. We don't acquire numeracy, we acquire innumeracy. If we practiced preventive medicine, we woudn't need to search for cures for innumeracy.
To function in an intuitive mode, that is to understand something in other than rote fashion, Dehaene claims the mind needs images, and math education should help children build a rich repertoire of "mental models" of arithmetic. Hadamard mentions the predominance of images in the creative thought of mathematicians. So, manipulatives are not to be used and then discarded to be replaced by abstract thinking. They are to be used to create mental models that we can use to carry information and provide understanding. Our visual memory is very strong. For example, I have a very strong picture of a board foot, but no memory at all of a formula. When I think of a board foot I see a one foot length of a 1 x 12; or, equivalently, a one foot length of a 2 x 6. (A 1 x 6 is half of a 1 x 12 so a 10 foot 1 x 6 has 5 board feet; a 2 x 4 is two-thirds of a 2 x 6, so a 10 foot
2 x 4 has 2/3 of 10 or 6 and 2/3 board feet, and so on.) If I wanted a formula, I would have to derive that from my mental picture. That is true for lots of mathematical concepts. The purpose of the Math and The Mind's Eye materials is to build images.
We have talked about how emotion easily overwhelms rational thought. The most important thing is to recognize an emotion and let it be. In and of itself, an emotion is neither good nor bad, it simply is. You can deal with emotions in either a constructive or destructive manner. However, it is not constructive to deny them. In other words, you are permitted to hate math. Once a counselor and I team-taught a workshop for secondary teachers on working with math anxious students. One of the things we stressed was to acknowledge the anxiety that existed and not try to make it go away. We role played, one person professing anxiety about a mathematical topic and another responding in an empathic way, giving permission to feel anxious. The exercise was a failure; the teachers could not bring themselves to do that. They kept insisting that things would be okay, or kept trying to find the source of the anxiety. None of them were able to say, "Yes, I hear the anxiety in your voice, and it is okay with me if you are feeling anxious. I still want you to give this activity a go." The teachers were having a difficult time accepting the feelings being expressed and not taking responsibility for them.
Take a Break
I think all of us have experienced those unexpected flashes of cognition such as Poincare described—when that bit of information or a solution to a problem pops into our conscious mind after we have given up the conscious search. We don't know much about how it works, but it seems to follow a period of conscious effort followed by, in Hadamard's words, a period of incubation during which our attention is diverted elsewhere. However, I believe when we are getting nowhere in working something out, we can facilitate the problem-solving process by deliberately stopping our efforts. I make a conscious effort to do this. It is not easy because once I set my mind on trying to figure something out I don't want to let it go until I am successful. I like word puzzles, double-crostics, and cryptic crosswords. They give me an opportunity to practice letting go. Every morning I do the JUMBLE. It usually involves some rather atrocious pun. If I get stuck trying to figure out the pun, I try to put it aside, do something else and come back to it later. It amazes me how often this works. When I teach a class, rather than admonish students to work hard, I tell them, "If you are working on something for class and are not making progress, your assignment is to quit before you start to feel frustrated. Say to yourself, "I will know more about this when I come back to it later." Asking them to do this wards off feelings of frustration and messages like, "I'll never get this," or "I must be stupid," or "I hate this." My sense is that feeding these messages to the subconscious doesn't give it permission to keep thinking about the problem. All of this is speculation on my part, but it does create a more relaxed classroom and lots of interesting stories about when a good idea occurred.
As Dehaene points out, there are some things our brains aren't very good at. We can do them, but it takes effort. Computation is one of those things. We don't have a CPU that is devoted to computing, so we call on help from all over our brain. Memorizing isolated information is also difficult. We can do it in verbal memory but then we sacrifice meaning. To compensate for our difficulties we have developed technological solutions; as Sylwester puts it, adding an exterior technological layer to our brain. As far as computation is concerned, we have developed technological devices that aid us in the task of computing: counting boards, Napier's bones, abaci, paper and pencil, trig tables, manual adding machines, electric adding machines, electronic calculators, computers, and so forth. What amazes me is that we don't embrace them and accept the help they offer. What is particularly puzzling to me is how readily we accept a sixteenth century invention, the lead (actually graphite, which is a form of carbon) pencil, while denying the use of a twentieth century invention. I am reminded of my school days when I had to use a stick pen rather than a fountain pen (this was prior to ball points!), and wasn't allowed to type papers. Typing was only available to secretarial science students.
I think the problem is confusion about what is basic to learning and using. If I can have a career teaching math, doing research in mathematics, working in industry as a mathematician, and never have used the long division algorithm or anything based on it (as far as I know, there is absolutely no market for long division experts), how can it be a basic skill? Yet we devote countless education resources trying to get our students proficient at long division—at best a school survival skill. Similar things could be said about other algorithms and some of the other things we stress in school. For example the rapid recall of times tables. I think that as math educators we have to be very cautious in our approach to algorithmic learning. Teaching and drilling me on an algorithm for computing board feet before turning me loose in a lumberyard would have been a great disservice to me. What did put me in good stead was a good number sense and the old timer who showed me what a board foot was, and checked to see that I grasped the idea. In the course of a day I might have used a half dozen different methods for computing board feet, depending on what I was handling. Isn't that what we are striving for? Isn't that the most basic mathematical skill one can possess—a well developed intuitive number sense and arithmetical operations based on innate knowledge that enables one to develop his own arithmetical procedures as the demand arises?