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Why is Math So Hard?

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Barbara Dautrich teaches Teaching Math in the Elementary and Early Childhood Classroom for prospective educators.

Do you have frightening recollections of the classic math problem in which two trains depart from different locations traveling at different speeds? Did you panic trying to figure out the exact time they would meet in Chicago? Like the rest of us, you would rather have taken a bus.

It seems that almost no one escapes the pains and pitfalls of learning math, and yet experts claim that learning math can be fun and easy. They suggest that if you fear math, it is probably due to grade school trauma at the hands of a heartless teacher who didn’t like math. Although there is certainly some unfortunate history regarding how math has been taught in the past, it is simplistic to attribute all of the challenges of math to a single cause.

So, why is math so hard? To find the answers, educators have turned to the expertise of neuroscientists who conduct brain imaging using the latest functional magnetic resonance imaging (fMRI) technology. This allows them to trace the neural pathways of the thinking brain “in action.”

It appears that learning math, perhaps more than other subject areas, involves a widespread interplay of brain activity in which many brain mechanisms must be simultaneously activated and synchronized. Concept formation, memory, and processing speed for mathematical thinking can vary widely from individual to individual depending on the efficiency of these mechanisms. Because math demands such widespread activity in the brain, there are several key processes that are potential trouble spots.

The Role of Language in Learning Math Facts and Computation

Paradoxically, it is the language centers of the left hemisphere that are linked most directly to mathematical rote learning and factual memory. Brain studies show that learning facts and discrete pieces of knowledge activate the neural mechanisms of short-term memory in the hippocampus, a cauliflower-looking structure deep in the brain’s center. The hippocampus serves to keep current information “active.”

The rehearsal of information in short-term memory, like when a learner practices math facts or computations, appears to be critical in establishing permanent memory in the long-term stores of the brain. In math, basic knowledge becomes consolidated in long-term memory, and math facts and procedures gradually become automatic. It is exactly this automaticity that “frees up” the processing load on the brain so it can dedicate more mental energy to the higher-order tasks of reasoning and complex problem solving.

The language area also assists in regulating the step-by-step procedures needed for computation problems, such as division of large numbers. It appears that the information needed for the brain’s internal “calculator” is mediated through the verbal channels of the language system. You may notice the tendency to “talk to yourself” or move your lips during mathematical concentration as evidence of the key role of language. The synchronization of left-hemispheric activity is vital to successful problem solving.

Cognitive psychologists, who study the brain’s role in thinking, learning, and memory, explain how these factors come into play. For example, if a word problem requires a student to calculate the total cost of three CDs offered at a 25% discount, the language area of the brain makes sense of the words and then “holds” the verbal and numerical information in short-term memory while other mechanisms retrieve the facts and procedures needed for computation that are stored in long-term memory. The neural integration of short-term and long-term functions during complex learning is more simply called “working memory.”

Individuals who struggle with math may feel that they simply lack the ability to remember and recall the explicit parts of math. It is not uncommon to find that children who struggle to learn math have deficits in working memory. They may rely on finger-counting strategies longer than most, have difficulty with their addition and times tables, struggle to carry out mental math (doing math in one’s head), and find learning rules, formulae, and sequences for math troublesome. Adults with these kinds of math difficulties may have problems making or counting change, balancing a checkbook, itemizing budgets, keeping score, remembering dates and times, following the steps for computation, and keeping focused during math tasks.

Cognitive psychologists report that for these individuals, working memory can be easily disrupted by problems with attention, emotional problems, learning disabilities, and general math anxiety. In extreme cases, these difficulties may lead to phobia of mathematics and fear of mathematical devices.

The Role of Intuition and Reasoning in Math

Knowing your math facts and mastering procedural skills is only part of the picture when it comes to being good at math. Numerical reasoning, concept formation, and an intuitive sense of “how math works” are equally fundamental.

Great mathematicians, including Albert Einstein, have reported that math is mostly intuitive and that they could “see” the mathematical relationships and solutions in their minds. It is the non-verbal, sensory, and visual-spatial areas associated with the right hemisphere that are dedicated to this “fuzzier side” of math. There is evidence that some math abilities are inborn in the human brain. Researchers report that even young infants are able to discern elements of quantity, size, and shape. For many people who “get” math, there seems to be a helpful reliance on the non-verbal processes of the brain.

For example, if you are planning a picnic, you will visualize the size of the group, the seating space you will need, the estimated quantity of food you will need, and the approximate expense to be incurred. It is the job of the non-verbal areas of the brain to estimate amounts; perceive patterns; make sense out of relative attributes of size, shape, and time; and then carry out mathematical activity that “makes sense.”

In particular, the visual cortex and the cerebellum, located in the back of the brain, are linked to the spatial and directional processes used to “image” mathematical relationships. Specific structures in the parietal lobe on the brain’s upper surface help create perceptions of “how many” and “how much.” These structures also assist in knowing which operation to use (addition, division, etc.) and in manipulating numerical quantities in ways that make sense. The sensory cortex, a strip that lies across the top of the brain, is responsible for interpreting concrete, sensory experiences and helps give us that “feel” for mathematical thinking.

Because of the new knowledge of how the brain works, today’s classrooms are better geared to provide rich, hands-on math activities that generate a lot of input for the developing brain. You may find children working together at tables or on the floor, using colorful cubes, counters, or other objects to learn how multiplication and division are related. Others may be using a fold-out number line on the floor to hop through basic number facts. What looks like play is actually important multisensory experience for understanding how math works.

People who have problems with the intuitive, non-verbal processes of math may have difficulty with such things as doing math in their heads (mental math); judging time and the passage of time; reading maps; doing jigsaw puzzles; sensing direction; estimating distance, time, and amounts; understanding physical configurations of objects; converting units of measure (inches to feet, cups to quarts); and understanding how equations and formulae work.

Putting It All Together

Those who are gifted in math appear to be born with an innate talent for math along with an excellent memory for facts and numbers. The brain images of gifted math students show superior activation of the right-hemispheric functions and heightened interaction between the two hemispheres of the brain that speed up the synchronized exchange of information. The contributions of the left and right hemispheres are so well integrated that the brain acts uniformly and math becomes “whole.”

But for many of the rest of us, math seems hard because the “multitasking” of the brain is not perfectly automatic or efficient. We juggle the numbers in short-term memory, trying to figure out the best way to proceed in order to get an answer that will make sense. Most neural multitasking happens below conscious awareness and at speeds that are merely fractions of a second. So when the process breaks down, it can be hard to slow math down, bring it under conscious control, and rethink the problem.

Trying to form intuitive insight under “slow motion” conditions is especially vulnerable to the effects of anxiety and fluctuations in attention and motivation. Fear of failure and poor performance can become self-perpetuating.

What Can Be Done to Help

Fortunately, learning math facts and computation generally respond favorably to rote practice and repetition. For elementary students, practice should be periodic and for short intervals rather than tackled in marathon sessions. It is best to review what a child “does” know and gradually add in new items or levels. Using counting chips or beans can help a child visualize the numerical relationship being mastered; understanding the math concept should always precede memorization. Using fingers or counting-on strategies should not be discouraged during the learning phase.

For young children, early exposure to numbers and real-life counting tasks is important. Wise parents take natural opportunities to model rudimentary adding and subtracting tasks using toys, cookies, etc. Lots of language and hands-on, sensory experiences foster early math learning.

For older students and adults who do not achieve math fluency, calculators are indispensable. The goal at this stage is to find meaningful solutions to real problems as they are needed. The calculator itself can be an effective learning tool, displaying addition or multiplication tables at the repeated touch of a button, decimals and percents using multipliers of 10 or 100.

But perhaps the greatest challenge of learning math is trying to grasp those elusive, intuitive processes needed for understanding how math works. Because the capability for math intuition is thought to be inborn, some experts suggest that it cannot be directly taught. Instead, it seems that math insights are “acquired” incidentally from physical and sensory experiences in the real world.

Parents and teachers are advised that teaching math should extend from a child’s natural exploration of the environment. For example, children typically learn how time is structured into hours, days, and weeks by participating in family life and the everyday world. Concepts like quantity, size, distance, and time are similarly learned through common tasks such as shopping, eating, traveling, etc. The vocabulary of conceptual math abounds in everyday usage; words like “more,” “next,” “together,” “left over,” “after,” and “different” are all important math concepts that ordinary experience provides.

For adults, using everyday math at your own comfort level is the best way to stay sharp and pick up new skills. Notice the math around you when shopping for the best sale prices, at the gym when you walk a 20-minute mile, or when you estimate a 15% tip. Let the numerical concepts work in your mind until you “see” how the numbers make sense. Play with number relationships. When will you be exactly twice as old as your child? How much sooner will you arrive if you go 60 miles per hour instead of 50 miles per hour? How much is it costing you per day to feed your pet(s)? Since these mini exercises are connected to real experiences and not high-stakes math (like balancing a checkbook), you can gradually improve your skills while reducing your avoidance of math.

With just a glimpse of the remarkable inner workings of the brain, we are reminded that the human capacity for thinking, learning, and memory is nothing short of miraculous. Although calculating the arrival of speeding trains may leave us feeling “derailed,” there is no question that the mathematical brains we take for granted serve us amazingly well in our daily routines and busy lives.

About the Author

Barbara Dautrich, Ed.D., is director of the elementary education program at American International College in Springfield, MA. She has published research articles on educational psychology, as well as a nationally published study guide on the subject. She received her bachelor’s degree from Westfield State College and her master’s and doctoral degrees from AIC.
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