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Steve Wyborney's Blog: I'm on a Learning Mission.

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Steve Wyborney's Blog: I'm on a Learning Mission. - Sharing Thoughts About Education

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Division:  A Remainder of One interactive read aloud

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Teaching Preschool and Kindergarten Math: More Than 175 Ideas, Lessons, and Videos for Building Foundations in Math, A Multimedia Professional Learning Resource

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Through an exciting multimedia format, Teaching Preschool and Kindergarten Math takes you into an early childhood classroom for a seeing is believing look at how to create a focused, successful mathematics program while simultaneously deepening your knowledge of the mathematical ideas that need t...

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Bridges in Mathematics | The Math Learning Center | MLC

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The Math Learning Center offers a comprehensive standards-based math program as well as innovative supplemental resources.

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I train mathematics educators who want their students to be better problem solvers.

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Use these five strategies to encourage meaningful classroom discussions.

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Teaching Tools | Resources for Teachers from Scholastic

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At youcubed, we inspire, educate and empower teachers of mathematics, transforming the latest research on mathematics learning into accessible and practical forms.

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How your brain processes complex mathematical problems is very different from how it processes language.

Credit: ChristianChan

Albert Einstein once said that his mathematical genius had nothing to do with language: "Words and language, whether written or spoken, do not seem to play any part in my thought processes."

And now high-definition brain scans prove he had a point. The ways that the brain processes language and complex mathematical concepts are quite different, according to a new study.

The notion that humans first developed mathematical abilities as an offshoot of early forms of language has been a long-standing hypothesis, according to the study authors. And some studies have suggested that the way the brain wrangles abstract math concepts has more in common with language processing than it does with simple number recognition and formula computing. [ 3D Images: Exploring the Human Brain]

But this idea has opponents - including many mathematicians, the researchers noted - who argue that understanding complex math involves perception pathways that differ greatly from those that untangle words.

To find out which idea held true, researchers turned to a type of noninvasive scan called fMRI (functional magnetic resonance imaging), which creates high-resolution images that can map changes in neural activity triggered by blood flow. In that way, researchers would be able to see which parts of the brain lit up during different types of tasks.

For the experiments in the study, the scientists selected 15 subjects who were trained mathematicians and 15 subjects who were well educated but did not specialize in math. The researchers posed a series of statements to each subject on a range of topics - math and nonmath - which participants were asked to identify as "true," "false" or "meaningless."

When both groups responded to statements on subjects that had nothing to do with math, the brain scans showed activity in the regions associated with language processing.

But when the mathematicians pondered and responded to statements relating to advanced mathematical concepts, certain regions of their brains - the prefrontal, parietal, and inferior temporal regions - lit up. Those brain regions also activated in nonmathematicians only when they were puzzling over numbers and formulas. During this activity, the researchers found, did not occur at all in brain regions linked to words.

According to the study's co-author Marie Amalric, their findings indicate that understanding math on a conceptual level uses the same neural networks that recognize numbers and space, rather than the networks that process general knowledge.

Amalric, a neuroscientist with the Cognitive Neuroimaging Unit at the NeuroSpin Center in France, told Live Science in an email that prior studies suggested that an early childhood ability to recognize numbers could possibly predict aptitude for performing more abstract mathematical reasoning later in life.

"Such applications are a bit far from our conclusions," Amalric said. "However, our findings are in accordance with previous conclusions showing that the number sense is a good predictor of further mathematical skills."

The study was published online today (April 11) in the journal Proceedings of the National Academy of Sciences.

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Mathematical Ability Revealed in Brain Scans

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Albert Einstein once said that his mathematical genius had nothing to do with language: "Words and language, whether written or spoken, do not seem to play any part in my thought processes." And now high-definition brain scans prove he had a point.

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Processing high-level math concepts uses the same neural networks as the basic math skills a child is born with

By Jordana Cepelewicz

Alan Turing, Albert Einstein, Stephen Hawking, John Nash-these "beautiful" minds never fail to enchant the public, but they also remain somewhat elusive. How do some people progress from being able to perform basic arithmetic to grasping advanced mathematical concepts and thinking at levels of abstraction that baffle the rest of the population? Neuroscience has now begun to pin down whether the brain of a math wiz somehow takes conceptual thinking to another level.

Specifically, scientists have long debated whether the basis of high-level mathematical thought is tied to the brain's language-processing centers-that thinking at such a level of abstraction requires linguistic representation and an understanding of syntax-or to independent regions associated with number and spatial reasoning. In a study published this week in Proceedings of the National Academy of Sciences, a pair of researchers at the INSERM-CEA Cognitive Neuroimaging Unit in France reported that the brain areas involved in math are different from those engaged in equally complex nonmathematical thinking.

The team used functional magnetic resonance imaging (fMRI) to scan the brains of 15 professional mathematicians and 15 nonmathematicians of the same academic standing. While in the scanner the subjects listened to a series of 72 high-level mathematical statements, divided evenly among algebra, analysis, geometry and topology, as well as 18 high-level nonmathematical (mostly historical) statements. They had four seconds to reflect on each proposition and determine whether it was true, false or meaningless.

The researchers found that in the mathematicians only, listening to math-related statements activated a network involving bilateral intraparietal, dorsal prefrontal, and inferior temporal regions of the brain. This circuitry is usually not associated with areas involved in language processing and semantics, which were activated in both mathematicians and nonmathematicians when they were presented with the nonmathematical statements. "On the contrary," says study co-author and graduate student Marie Amalric, "our results show that high-level mathematical reflection recycles brain regions associated with an evolutionarily ancient knowledge of number and space."

Previous research has found that these nonlinguistic areas are active when performing rudimentary arithmetic calculations and even simply seeing numbers on a page, suggesting a link between advanced and basic mathematical thinking. In fact, co-author Stanislas Dehaene, director of the Cognitive Neuroimaging Unit and experimental psychologist, has studied how humans (and even some animal species) are born with an intuitive sense of numbers-of quantity and arithmetic manipulation-closely related to spatial representation. How the connection between a hardwired "number sense" and higher-level math is formed, however, remains unknown. This work raises the intriguing question of whether an innate capability to recognize different quantities-that two pieces of fruit are greater than one-is the biological foundation on which can be built the capacity to master group theory. "It would be interesting to investigate the causal chain between lower-level and higher-level mathematical competency," says Daniel Ansari, a cognitive neuroscientist at the University of Western Ontario who did not participate in the study. "Most of us master basic arithmetic, so we're already recruiting these brain regions, but only a fraction of us go on to do high-level math. We don't yet know whether becoming a mathematical expert changes the way you do arithmetic or whether learning arithmetic lays out the foundation for acquiring higher-level mathematical concepts."

Ansari suggests that a training study, in which nonmathematicians are taught advanced mathematical concepts, could provide a better understanding of these connections and how they form. Moreover, achieving expertise in mathematics may affect neuronal circuitry in other ways. Amalric's study found that mathematicians had reduced activity in the visual areas of the brain involved in facial processing. This could mean that the neural resources required to grasp and work with certain math concepts may undercut-or "use up"-some of the brain's other capacities. Although additional studies are needed to determine whether mathematicians actually do process faces differently, the researchers hope to gain further insight into the effects that expertise has on how the brain is organized.

"We can start to investigate where exceptional abilities come from, and the neurobiological correlates of such high-level expertise," Ansari says. "I just think it's great that we now have the capability to use brain imaging to answer these deep questions about the complexity of human abilities."

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How Does a Mathematician's Brain Differ from That of a Mere Mortal?

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Processing high-level math concepts uses the same neural networks as the basic math skills a child is born with

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This is Mathematics Awareness Month, and this year's theme is Mathematics and the Brain...

Joe Malkevitch
York College (CUNY)
malkevitch at york.cuny.edu

Introduction

The volume of a typical human brain is 1400 cm ³ and its weight is about 1000 times that of a typical rat's brain. Mundane facts about brain data do little to give us insight into this amazing organ. What is there about the human brain that seems to make possible so many wondrous things that humans can accomplish: oral and written language, producing of art and music, and doing mathematics. Or is it really possible that dolphins have already resolved the status of the Riemann Hypothesis but we are unaware of it because we have found no way yet to communicate with them?

As so often seems to happen, at first glance there might not seem to be much to say about this topic. However, there are more facets to this subject than I can treat here or know enough about to say anything intelligent about them. I will merely try to provide you with a "wire" to connect mathematics and brain.

Imaging the Brain

One of the reasons the brain was a mystery for so long is that it is encased in the skull. Under ordinary circumstances one does not have easy access to seeing the brain. Being able to see into the human body took a big leap forward when Wilhelm Roentgen discovered x-rays. Shortly after his discovery of x-rays physicians were already using x-rays to see our bones. However, being encased by the skull, the brain, and other soft tissues were still unknown territory for physicians and physiologists. With contributions from science, engineering and mathematics, physicians have a growing array of imaging tools for seeing inside the skull. At first these brain imaging techniques involved 2-dimensional cross sections of the brain. Many of these techniques are identified with the field of tomography. The Greek word for a "cut" or "section" is "tomas." ( Tomography has applications in fields other than medicine., for example, geology.) As the field of tomography has matured, it has been possible to generate three-dimensional medical imaging systems for the brain. These systems allow static views of the brain; increasingly it is also possible to make movies that show the dynamic behavior of the brain. The systems that are now available both to medical practitioners and researchers are:

* CT or CAT ( Computed tomography or computed axial tomography)

* MRI (Magnetic Resonance Imaging)

* fMRI ( Functional MRI)

* PET ( Positron Emission Tomography)

* SPECT (Single Photon Emission Computed Tomography)

* DOT (Diffuse Optical Tomography)

"Vanilla" samples of some of these types of images are shown below:

Images of the kind shown above have become part of the lexicon of our society. They show the brain using three different techniques that physicians have come to rely upon for insight and diagnosis.

The development of the CT-scans, computerized tomography, has proved to be one of the great leaps forward in diagnostic medicine and for medical research. It is a also a wonderful story of the interplay of pure mathematics, applicable mathematics, and computer science.

The story, amazingly enough, starts in Czechoslovakia. The mathematician Johnann Radon (1887-1956) was investigating the idea of reconstructing the value of a function at a point in a plane region by computing the sums (integral) of the values of this function along lines that intersected this domain. This work was published in German in 1917. Radon's work evolved into what is known today as the Radon Transform.

It is unclear if Radon saw a future for a medical revolution in his work but in any case, the engineering and the computer science to implement the implications of his ideas were in the future. Furthermore, his mathematical ideas also had to be amplified to make practical tomography a reality.

You can get the flavor of Radon's idea by consider the example below which involves a discrete analog of the question Radon investigated.

The vertices of the "square head" have unknown density, represented by x, y, z, and w. If one knows the sums of these unknown values along some (all) of the 6 line segments that are formed by the vertices of the square, can one reconstruct the unknown densities?

For example, using 4 of the 6 lines suppose we get the equations:

x + y = 8

w + z = 10

y + z = 13

w + y = 7.

Readers who know a little linear algebra may wish to see what happens for the general case here. Readers who know high school algebra should be able to determine that the equations above have a unique solution.

Usually four equations in four unknowns will give rise to a unique solution. However, it might be that these 6 equations are inconsistent and have no solution, or the solution might depend on which of the 4 out of 6 equations one selects. Problems of this kind belong to the emerging field of " discrete tomography." Such problems (the problem above involves the 2-dimensional cube), like Radon's original work, belong to the area of reconstruction of information from partial knowledge, or from "samples" of various kinds.

Radon showed that the unknown densities could be reconstructed from the "line integrals" that he was computing. Whether he or anyone realized the value of Radon's work for medical imaging at the time is unclear, but as noted previously, without significant computer power Radon's work could not be put to use immediately.

The story of CT scans picks up in more modern times with the work of Godfrey Hounsfield (1919-2004) and Allan Cormark (1924-1998). Hounsfield was an English electrical and mechanical engineer and Cormak was a South African physicist. The two men shared the Nobel Prize for Physiology and Medicine in 1979 for their work. However, in initially implementing the creation of a scanner in 1971 (based on earlier ideas) no use was made of Radon's work. Larry Shepp, a mathematician at Rutgers University, heard a talk by Hounsfield in which he described his new device. It emerged that Hounsfield was unaware of the possibility of using Radon's work in conjunction with his new machine, and Shepp became involved in developing computer code to improve the image quality for the new technology.

As time has gone on, mathematicians have become increasingly involved with imaging technologies of many kinds.

The human visual system

One burgeoning area connecting mathematics and the brain is visual phenomena. Human visual information is processed by the brain and uses our amazing eyes. Mathematicians have been involved in a wide variety of problems involving vision:

* What is the geometry of our perceptual system?

* How does the brain process visual information?

It is well known that when humans look at (straight) railroad tracks they appear to converge in the distance. A wide range of scholars are trying to understand the mystery of visual perception. The way that the world looks to us was a challenge for artists who had to represent reality on canvas, and for philosophers who are trying to probe the relation between mind, physics, and perception. Artists met this challenge with a theory of perspective and mathematicians (some of them also artists) transformed the insights obtained concerning perspective into what today is known as projective geometry. In projective geometry, unlike Euclidean geometry, there are no parallel lines. However, this is not where the story ends. It has been suggested that the hyperbolic plane or other geometries might describe this system. The challenge of understanding perspective from a mathematical point of view and being able to use computers to serve the needs of artists, photographers, and architects has been a source of much research.

Illusions and impossible figures

One interesting connection between mathematics and the brain is the reaction many people have to symmetric patterns and patterns which are deceptive ( visual illusions and impossible figures). Many individuals including mathematicians find symmetrical designs very appealing. For example, the works of M.C. Escher (1898-1972) hit a very sympathetic chord in the mathematics community.

Furthermore, these is a connection between Escher and interest in optical illusions and "impossible objects." This link is via Roger Penrose (1931-), the distinguished mathematician and physicist, whose impossible triangle has inspired a large number of artistic efforts, including those of Escher, as well as mathematical and computer vision investigations.

However, it not as widely known as it should be that Penrose was not the first to create such images, though his version of the illusion is particularly appealing.

In the diagram above the version due to Roger Penrose is shown on the right but the illustration on the left shows a version of this object first created in 1934 by the Swedish artist Oscar Reutsvärd. Reutsvärd's work has been commemorated by a Swedish stamp and his version of the impossible triangle is but one of many such illusions that he created. It turns out that another aspect of "impossible figures" is making real objects which look exactly like impossible ones. There is more to illusions than meets the eye or the brain.

Illusions open up a window to explore the nature of our visual system, the way the brain processes visual information, and the geometry of our visual perception system. Experts in computer vision, a subject which includes effective ways of manipulating images by computers as well as developing software to carry out various pattern recognition problems, are also interested in visual illusions. Applications of work in this area might range from more effective face recognition software to scene recognition capability for software. Here is a small sample of visual illustions which show some the range of phenomena that might be explored concerning why the brain is "fooled" and which can perhaps be modeled using mathematics.

Other mathematical windows on the brain

The small sample of ways that mathematics interacts with our knowledge of the human brain barely hints at the myriad of interfaces of these two subjects. What follows is just an additional sampler of ways that mathematics is interacting with and providing a valuable tool for attempts to understand the brain.

* Neuronal organization and transmission of signals (nerve conduction) between the brain and other parts of our nervous system

* The use of dynamical systems in an attempt to understand epilepsy and to be able to try to predict the start of seizures

* Modeling various aspects of the brain using mathematics which tries to understand how "oscillators" coordinate their firings under various interaction assumptions

* Using new ideas about flattening complicated surfaces (like the brain) to understand how to map different parts of the brain to specific brain functions

* Using data mining and other tools to understand the implications of an increasing wealth of DNA and genomic information for brain chemistry

Mathematical ability

To the best of our knowledge humans are the only species that does mathematics. Attempts to verify that other animals (plants?) can "do" mathematics have so far not encouraged the belief that any mathematics is being produced by other species. The famous story of " clever Hans," a horse whose master had convinced himself that Hans could do mathematics turned out to be not a case of a horse with mathematical talent but a human master who was sending signals to the horse without realizing he was doing this. What is unique about the human brain that makes this doing mathematics possible? Brain imaging techniques are being used to chart the differences in how humans might do calculations, visualize a polyhderon, try to prove a theorem, or brain storm a problem in number theory. What are the ways that having a "good memory" makes mathematics easier? Many mathematicians describe themselves as having a good memory. What can be said is that mathematicians are using their brains and mathematics to further mathematics' domains and to get further insight into how mathematics gives insight into brains!

References

Anderson, S., and D. Burr, Spatial and temporal selectivity of the human motion detection system, Vision Research 25 (1985) 1147-1154.

Anger, B., and C. Portenier, Radon Integrals, Birkhauser, Boston, 1992.

Babloyantz, A., and C. Lourenco, Brain chaos and computation. Int J Neural Syst 7 (1996) 461-71.

Beigler, R. and A. McGregor, and S. Healy, How do animals "do" geometry?, Animal Behavior 57 (1999) F4-F8.

Blakemore, C., and F. Campbell, On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images, J. of Physiology (1969) 203-237.

Cabeza, R., and A. Kingstone, (eds.), Handbook of Functional Neuroimaging Cognition, second edition, MIT Press, Cambridge, 2006.

Chomsky, N., Language and Mind, Harcourt Brace Jovanovich, New York, 1972.

Cowan, T., The theory of braids and the analysis of impossible figures, J. Math. Psychology 11 (1974) 190-212.

Cowan, T., Turning a Penrose triangle inside out, J. Math. Psychology 26 (1982) 252-262.

Cowan, T., and R. Pringle, An investigation of the cues responsible for figure impossibility, J. of Experimental Psychology: Human Performance and Perception, 4 (1978) 112-120.

Deans, S., The Radon Transform and Some of Its Applications, Wiley, New York, 1968.

De Cruz, H., Towards a Darwinian approach to Mathematics, Foundations of Science, 11 (2006) 157-196.

Dehene, S., Cerebral bases of number processing and calculation, In M. Gazzaniga (ed.), The New Cognitive Neurosciences, (Second Edition), MIT Press, Cambridge, 2000, p. 987-998.

Derchansky, M., Examining the Epileptic Brain with Nonlinear Mathematics (Preprint).

De Valois, R., and K. De Valois, Spatial Vision, Oxford Press, Oxford, 1988.

Esser, P. (ed.), Emission Computed Tomography: Current Trends, Society of Nuclear Medicine, New York, 1983.

Gardner, R., Geometric Tomograpy, Cambridge U. Press, New York, 1995.

Gindikin, S. (ed.), Applied Problems of Radon Transforms, American Mathematical Society, Providence, 1994.

Greenfield, S., The Human Brain: A Guided Tour, Phoenix, London, 1998.

Griffin, D., Animal Minds. Beyond Cognition to Consciousness, U. of Chicago Press, Chicago, 2001.

Grünbaum, B. and G. Shephard, Tilings and Patterns, W.H. Freeman, New York, 1987.

Helgason, S., The Radon Transform, Birkhauser, Boston, 1980.

Herman, G., Image Reconstruction from Projections: The Fundamentals of Computerized Tomography, New York, Academic Press, 1980.

Herman, G. and A. Kuba (eds.), Discrete Tomography: Foundations, Algorithms, and Applications, Birkhauser, Boston, 1999.

Hoffman, W., Subjective geometry and geometric psychology, Mathematical Modeling 1 (1980) 349-367.

Hong, W., A. Yang, K. Huang, Y. Ma, On symmetry and multiple-view geometry: Structure, pose, calibration from a single image, International J. of Computer Vision 60 (2004) 241-265.

Hoppensteadt, F., An Introduction to the Mathematics of Neurons, Cambridge U. Press, New York, 1986.

Huffman, D., Impossible objects as nonsense sentences. In B. Meltzer and D.
Lorenceau, J., Geometry and the visual brain, J. of Physiology, 97 (2003) 99-103.

Michie (eds.), Machine Intelligence, Volume 6, Halstead, New York, 1971.

Lehnertz, K., Non-linear time series analysis of intracranial EEG recordings in patients with epilepsy--an overview., Int J Psychophysiol 1999;34(1):45-52

Lehnetz, K., and C. Elger, Can epileptic seizures be predicted? Evidence from nonlinear time series analysis of brain electrical activity, Phys Rev Lett 1998;80:5019-26.

Le Van Quyen, M., Anticipating epileptic seizures: from mathematics to clinical applications, Comptes Rendus Biologies 328 (2005) 187-198.

Li, S., and D. Mital, E. Teoh, H. Wang, (eds.), Recent Developments in Computer Vision, Springer, Berlin, 1995.

Ma., Y., J. Kosecka, S. Soatto, S. Sastry, An Invitation to #-D Vision: From Images to Geometric Models, Springer-Verlag, New York, 2003.

Penrose, L., and R. Penrose, Impossible objects: A special type of illusion, British J. of Psychology, 49 (1958) 31-33.

Plamer, S., Vision Science: Photons to Phenomenology, MIT Press, Cambridge, 1999.

Rann, A. and A. Katsevich, The Radon Transform and Local Tomography, CRC, Boca Raton, 1996.

Shepp, L., (ed.), Computed Tomography, Volume 27, Proceedings of Symposia in Applied Mathematics, American Mathematical Society, Providence, 1983.

Shepp, L., and J. Kruskal, Computerized tomography: The new medical X-ray technology, Amer. Math. Monthly 85 (1978) 420-439.

Starkey, P., and R. Cooper, Perception of Numbers by Human Infants, Science 210 (1980) 1033-35.

Sugihara, K., Machine Interpretation of Line Drawings, MIT Press, Cambridge U. Press,

Wynn, K., Addition and subtraction by human infants, Nature 358 (1992) 749-750.

Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some of these materials. Some of the items above can be accessed via the ACM Portal, which also provides bibliographic services.

Joe Malkevitch
York College (CUNY)
malkevitch at york.cuny.edu

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This is Mathematics Awareness Month, and this year's theme is Mathematics and the Brain... Joe Malkevitch York College (CUNY) malkevitch at york.cuny.edu The volume of a typical human brain is 1400 cm 3 and its weight is about 1000 times that of a typical rat's brain.

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The Learning and the Brain Conference unites neuroscientists, psychologists, and educators to explore the brain and the teaching-learning process. Here's what I learned.

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Erikson's psychosocial theory of development considers the impact of external factors, parents and society on personality development from childhood to adulthood. According to Erikson's theory, every person must pass through a series of eight interrelated stages over the entire life cycle[2].

1. Infancy: Birth-18 Months Old

Basic Trust vs. Mistrust - Hope

During the first or second year of life, the major emphasis is on the mother and father's nurturing ability and care for a child, especially in terms of visual contact and touch. The child will develop optimism, trust, confidence, and security if properly cared for and handled. If a child does not experience trust, he or she may develop insecurity, worthlessness, and general mistrust to the world.

2. Toddler / Early Childhood Years: 18 Months to 3 Years

Autonomy vs. Shame - Will

The second stage occurs between 18 months and 3 years. At this point, the child has an opportunity to build self-esteem and autonomy as he or she learns new skills and right from wrong. The well-cared for child is sure of himself, carrying himself or herself with pride rather than shame. During this time of the "terrible twos", defiance, temper tantrums, and stubbornness can also appear. Children tend to be vulnerable during this stage, sometimes feeling shame and and low self-esteem during an inability to learn certain skills.

3. Preschooler: 3 to 5 Years

Initiative vs. Guilt - Purpose

During this period we experience a desire to copy the adults around us and take initiative in creating play situations. We make up stories with Barbie's and Ken's, toy phones and miniature cars, playing out roles in a trial universe, experimenting with the blueprint for what we believe it means to be an adult. We also begin to use that wonderful word for exploring the world-"WHY?"

While Erikson was influenced by Freud, he downplays biological sexuality in favor of the psychosocial features of conflict between child and parents. Nevertheless, he said that at this stage we usually become involved in the classic "Oedipal struggle" and resolve this struggle through "social role identification." If we're frustrated over natural desires and goals, we may easily experience guilt.

The most significant relationship is with the basic family.

4. School Age Child: 6 to 12 Years

Industry vs. Inferiority - Competence

During this stage, often called the Latency, we are capable of learning, creating and accomplishing numerous new skills and knowledge, thus developing a sense of industry. This is also a very social stage of development and if we experience unresolved feelings of inadequacy and inferiority among our peers, we can have serious problems in terms of competence and self-esteem.

As the world expands a bit, our most significant relationship is with the school and neighborhood. Parents are no longer the complete authorities they once were, although they are still important.

5. Adolescent: 12 to 18 Years

Identity vs. Role Confusion - Fidelity

Up until this fifth stage, development depends on what is done to a person. At this point, development now depends primarily upon what a person does. An adolescent must struggle to discover and find his or her own identity, while negotiating and struggling with social interactions and "fitting in", and developing a sense of morality and right from wrong.

Some attempt to delay entrance to adulthood and withdraw from responsibilities (moratorium). Those unsuccessful with this stage tend to experience role confusion and upheaval. Adolescents begin to develop a strong affiliation and devotion to ideals, causes, and friends.

6. Young adult: 18 to 35

Intimacy and Solidarity vs. Isolation - Love

At the young adult stage, people tend to seek companionship and love. Some also begin to "settle down" and start families, although seems to have been pushed back farther in recent years.

Young adults seek deep intimacy and satisfying relationships, but if unsuccessful, isolation may occur. Significant relationships at this stage are with marital partners and friends.

7. Middle-aged Adult: 35 to 55 or 65

Generativity vs. Self absorption or Stagnation - Care

Career and work are the most important things at this stage, along with family. Middle adulthood is also the time when people can take on greater responsibilities and control.

For this stage, working to establish stability and Erikson's idea of generativity - attempting to produce something that makes a difference to society. Inactivity and meaninglessness are common fears during this stage.

Major life shifts can occur during this stage. For example, children leave the household, careers can change, and so on. Some may struggle with finding purpose. Significant relationships are those within the family, workplace, local church and other communities.

8. Late Adult: 55 or 65 to Death

Integrity vs. Despair - Wisdom

Erikson believed that much of life is preparing for the middle adulthood stage and the last stage involves much reflection. As older adults, some can look back with a feeling of integrity - that is, contentment and fulfillment, having led a meaningful life and valuable contribution to society. Others may have a sense of despair during this stage, reflecting upon their experiences and failures. They may fear death as they struggle to find a purpose to their lives, wondering "What was the point of life? Was it worth it?"

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Erikson's psychosocial theory of development considers the impact of external factors, parents and society on personality development from childhood to adulthood. According to Erikson's theory, every person must pass through a series of eight interrelated stages over the entire life cycle[2]. Basic Trust vs.

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The Scribble stage is made up of four sub-stages. (a) Disordered - uncontrolled markings that could be bold or light depending upon the personality of the child. At this age the child has little or no control over motor activity. (b) Longitudinal - controlled repetitions of motions.

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Imagine a world where intelligence is measured like this:

A child sits down at a desk. She is given a piece of paper and a crayon. Then, she is asked to draw a picture of a boy or girl. "Do the best that you can," she is told. "Make sure that you draw all of him or her." If the child hesitates, or asks for help, she is gently encouraged: "You draw it all on your own, and I'll watch you. Draw the picture any way you like, just do the best picture you can."

When the child is done drawing, the picture is scored. It's a simple process, with little ambiguity. One point is awarded for the "presence and correct quantity" of various body parts, such as head, eyes, mouth, ears, arms and feet. (Clothing gets another point.) The prettiness of the picture is irrelevant. Here are six drawings from four-year olds:

The Draw-A-Person test was originally developed by Florence Goodenough, a psychologist at the University of Minnesota. Based on her work with Lewis Terman - she helped revise and validate the Stanford-Binet I.Q. test - Goodenough became interested in coming up with a new measure of intelligence that could be given to younger children. And so, in 1926, she published a short book called The Measurement of Intelligence by Drawings which described the Draw-A-Person test.* Although the test only takes a few minutes, Goodenough argued that it provided a window into the child mind, and that "the nature and content of children's drawings are dependent primarily upon intellectual development." In other words, those scrawls and scribbles were not meaningless marks. Rather, they reflected something fundamental about the ways in which we made sense of the world. The act of expression was an act of intelligence, and should be treated as such.

In her book, Goodenough described the obvious benefits of her intelligence test. It was fast, cheap and fun. What's more, it seemed to be measuring something real, as children tended to generate a consistent set of scores over time. (In other words, the test was reliable.) And yet, despite these advantages, the Draw-A-Person test largely fell out of favor by the 1970s. One explanation is that it was lumped in with other "projective" techniques, such as the Rorschach Test, that were repeatedly shown to be inaccurate, too tangled up with psychoanalytic speculation.

However, a new study by Rosalind Arden and colleagues at King's College London suggests that Goodenough's test still has its uses, and that it manages to quantify something important about the developing mind in less than ten minutes. "Goodenough's genius was to take a common childhood product and see its potential as an indicator of cognitive ability," they write. "Our data show that the capacity to realize on paper the salient features of a person, in a schema, is an intelligent behavior at age 4. Performance of this drawing task relies on various cognitive, motoric, perceptual, attentional, and motivational capacities."

How'd the scientists show this? By giving the test to 7,752 pairs of British twins, the scientists were able to compare the drawing performance of identical twins, who share all of their genetic material, with that of non-identical twins, who only share about half. This allowed them to tease out the relative importance of genetics in determining scores on the Draw-A-Person test. (All of the twin pairs were raised in the same household, at least until age 4, so they presumably had a similar home environment.) The results were interesting, as the drawings of identical twins were much more similar than those of non-identical twins. There is no drawing gene, of course, but this result does suggest that the sketches of little kids are shaped by their genetic inheritance. In fact,the results from a single drawing were as heritable among the twin pairs as their scores on more traditional intelligence tests.

Furthermore, because the researchers had scores from these intelligence tests they were able to compare performance on the Draw-A-Person test with a subject's g factor, or general intelligence. The correlations were statistically significant but relatively modest, which is in line with previous studies. This means that one shouldn't try to predict IQ scores based on the scribbles of a toddler; the two variables are related, but in weak ways.

However, a more interesting result emerged over time, as the scientists looked at the relationship between drawing scores at the age of 4 and measures of intelligence a decade later, when the twins were 14. According to the data, the children's pictures were just as predictive of their intelligence scores at the age of 14 as various intelligence tests given at the age of 4. "This study does not explain artistic talent," write the scientists. "But our results do show that whatever conflicting theories adults have about the value of verisimilitude in early figure drawing, children who express it to a greater extent are somewhat brighter than those who do not."

Such studies trigger a predictable reaction in parents. I've got a three-year old daughter - I couldn't help but inspect her latest drawings, counting up the body parts. (There's even an app that will help you make an assessment.) But it's important to note that this is all nonsense; the science does not support my anxieties. "I too fossicked around in old drawers to look for body-parts among the fridge-magnet scrawls of my former 4-year old," Dr. Arden wrote in an email. "I realised quickly the key question was not 'is she bright?', but 'did we have fun? Did I treasure that wonderful, lightspeed flashing childhood properly?'" In a recent article put out by King's College, Arden expands on this idea, observing that while her "findings are interesting, it does not mean that parents should worry if their child draws badly. Drawing ability does not determine intelligence, there are countless factors, both genetic and environmental, which affect intelligence in later life."

I find this study most interesting as a history-of-science counter factual, a reminder that there are countless ways to measure human intelligence, whatever that is. We've settled on a particular concept of intelligence defined by a short list of measurable mental talents. (Modern IQ tests tend to focus on abilities such as mental control, processing speed and quantitative reasoning.) But Goodenough's tool is proof that the mystery of smarts has no single solution. The IQ test could have been a drawing test.

This sounds like a silly conjecture. But it shouldn't. As the scientists note, figurative art is an ancient skill. Before there were written alphabets, or counting systems, humans were drawing on the walls of caves. (There's evidence that children participated in these rituals as well, dragging their tiny fingers through the wet clay and soft cave walls.) "This long history endows the drawing test with ecological validity and relevance to an extent that is unusual in psychometrics," write the scientists. After all, the Make-A-Person test measures one of the most uniquely human talents there is: the ability to express the mind on the page, to re-describe the world until life becomes art, or at least a crayon stick figure.

*Goodenough originally called it the Draw-A-Man test, but later realized that the gendered description made it harder for young girls.

Arden, Rosalind, et al. "Genes Influence Young Children's Human Figure Drawings and Their Association With Intelligence a Decade Later." Psychological Science (2014)

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Imagine a world where intelligence is measured like this: A child sits down at a desk. She is given a piece of paper and a crayon. Then, she is asked to draw a picture of a boy or girl. "Do the best that you can," she is told.

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These stages are taken from Bob Steel's Draw Me a Story: An Illustrated Exploration of Drawing-as-Language, ©1997.

Scribble

This stage is typical of children between 18 months and 3 years.

Scribbles are random. Children are exploring art materials in a playful way.

Scribbles move from uncontrolled to progressively more controlled.

This stage allows children to learn to hold a pencil as well as to determine whether they are left or right-handed.

While you may not see it, some scribbles are named. The child will point to an object found in the scribble.

Scribbling helps children with:

Physical Development

Hand-eye coordination

Fine and gross muscle development

Hand manipulation

Language Art

Penmanship

Stroking

Finger dexterity

Fine muscle control

Guidance

Pre-Schematic Stage

This stage is typical of children between the ages of 2 to 4.

Drawings become more complex, although they are usually unrealistic.

Children will tend to use their favorite colors, rather than represent objects in accurate colors.

Drawings of people are very simple with few features.

Objects in drawings float in space. They are not anchored.

"Tadpole Figure People" are drawn with a very large head on a small body with extended arms.

Interiors and exteriors are shown at the same time. (X-Ray Drawings)

Children continue to develop increased hand-eye coordination, fine and gross muscle development, and self-confidence during this stage.

Observation

Thinking

Problem solving

Feelings of competence

Schematic Stage

This stage is typical of children between 5 to 8.

Drawings of people become more proportional and more detailed.

Colors become more realistic and stereotypical (grass is green, the sky is blue).

Skyline and ground lines start to show.

Children have a schema about a way of drawing. For example, a house will be drawn the same way in many drawings.

Children will often create stories to go along with their drawings.

Preteen Stage

This stage is typical of children between 9 to 11.

Drawings become far more detailed.

Much more spatial perspective is evident.

Children at this stage may become very frustrated if they are unable to create a realistic picture.

This is the time when children may express "I can't draw."

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Developmental Stages of Children's Drawings

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These stages are taken from Bob Steel's Draw Me a Story: An Illustrated Exploration of Drawing-as-Language, ©1997. This stage is typical of children between 18 months and 3 years. Scribbles are random. Children are exploring art materials in a playful way. Scribbles move from uncontrolled to progressively more controlled.

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March 2007 | Volume 64 | Number 6
Responding to Changing Demographics Pages 74-79

Robert J. Marzano and Debra J. Pickering

Homework is typically defined as any tasks "assigned to students by school teachers that are meant to be carried out during nonschool hours" (Cooper, 1989a, p. 7). A number of synthesis studies have been conducted on homework, spanning a broad range of methodologies and levels of specificity (see fig. 1). Some are quite general and mix the results from experimental studies with correlational studies.

Two meta-analyses by Cooper and colleagues (Cooper, 1989a; Cooper, Robinson, & Patall, 2006) are the most comprehensive and rigorous. The 1989 meta-analysis reviewed research dating as far back as the 1930s; the 2006 study reviewed research from 1987 to 2003. Commenting on studies that attempted to examine the causal relationship between homework and student achievement by comparing experimental (homework) and control (no homework) groups, Cooper, Robinson, and Patall (2006) noted,

With only rare exceptions, the relationship between the amount of homework students do and their achievement outcomes was found to be positive and statistically significant. Therefore, we think it would not be imprudent, based on the evidence in hand, to conclude that doing homework causes improved academic achievement. (p. 48)

The Case Against Homework

Although the research support for homework is compelling, the case against homework is popular. The End of Homework: How Homework Disrupts Families, Overburdens Children, and Limits Learning by Kralovec and Buell (2000), considered by many to be the first high-profile attack on homework, asserted that homework contributes to a corporate-style, competitive U.S. culture that overvalues work to the detriment of personal and familial well-being. The authors focused particularly on the harm to economically disadvantaged students, who are unintentionally penalized because their environments often make it almost impossible to complete assignments at home. The authors called for people to unite against homework and to lobby for an extended school day instead.

A similar call for action came from Bennett and Kalish (2006) in The Case Against Homework: How Homework Is Hurting Our Children and What We Can Do About It. These authors criticized both the quantity and quality of homework. They provided evidence that too much homework harms students' health and family time, and they asserted that teachers are not well trained in how to assign homework. The authors suggested that individuals and parent groups should insist that teachers reduce the amount of homework, design more valuable assignments, and avoid homework altogether over breaks and holidays.

In a third book, The Homework Myth: Why Our Kids Get Too Much of a Bad Thing (2006a), Kohn took direct aim at the research on homework. In this book and in a recent article in Phi Delta Kappan (2006b), he became quite personal in his condemnation of researchers. For example, referring to Harris Cooper, the lead author of the two leading meta-analyses on homework, Kohn noted,

A careful reading of Cooper's own studies . . . reveals further examples of his determination to massage the numbers until they yield something-anything-on which to construct a defense of homework for younger children. (2006a, p. 84)

He also attacked a section on homework in our book Classroom Instruction that Works (Marzano, Pickering, & Pollock, 2001).

Kohn concluded that research fails to demonstrate homework's effectiveness as an instructional tool and recommended changing the "default state" from an expectation that homework will be assigned to an expectation that homework will not be assigned. According to Kohn, teachers should only assign homework when they can justify that the assignments are "beneficial" (2006a, p. 166)-ideally involving students in activities appropriate for the home, such as performing an experiment in the kitchen, cooking, doing crossword puzzles with the family, watching good TV shows, or reading. Finally, Kohn urged teachers to involve students in deciding what homework, and how much, they should do.

Some of Kohn's recommendations have merit. For example, it makes good sense to only assign homework that is beneficial to student learning instead of assigning homework as a matter of policy. Many of those who conduct research on homework explicitly or implicitly recommend this practice. However, his misunderstanding or misrepresentation of the research sends the inaccurate message that research does not support homework. As Figure 1 indicates, homework has decades of research supporting its effective use. Kohn's allegations that researchers are trying to mislead practitioners and the general public are unfounded and detract from a useful debate on effective practice.

The Dangers of Ignoring the Research

Certainly, inappropriate homework may produce little or no benefit-it may even decrease student achievement. All three of the books criticizing homework provide compelling anecdotes to this effect. Schools should strengthen their policies to ensure that teachers use homework properly.

If a district or school discards homework altogether, however, it will be throwing away a powerful instructional tool. Cooper and colleagues' (2006) comparison of homework with no homework indicates that the average student in a class in which appropriate homework was assigned would score 23 percentile points higher on tests of the knowledge addressed in that class than the average student in a class in which homework was not assigned.

Perhaps the most important advantage of homework is that it can enhance achievement by extending learning beyond the school day. This characteristic is important because U.S. students spend much less time studying academic content than students in other countries do. A 1994 report examined the amount of time U.S. students spend studying core academic subjects compared with students in other countries that typically outperform the United States academically, such as Japan, Germany, and France. The study found that "students abroad are required to work on demanding subject matter at least twice as long" as are U.S. students (National Education Commission on Time and Learning, 1994, p. 25).

To drop the use of homework, then, a school or district would be obliged to identify a practice that produces a similar effect within the confines of the school day without taking away or diminishing the benefits of other academic activities-no easy accomplishment. A better approach is to ensure that teachers use homework effectively. To enact effective homework policies, however, schools and districts must address the following issues.

Grade Level

Although teachers across the K-12 spectrum commonly assign homework, research has produced no clear-cut consensus on the benefits of homework at the early elementary grade levels. In his early meta-analysis, Cooper (1989a) reported the following effect sizes (p. 71):

Grades 4-6: ES = .15 (Percentile gain = 6)

Grades 7-9: ES = .31 (Percentile gain = 12)

Grades 10-12: ES = .64 (Percentile gain = 24)

The pattern clearly indicates that homework has smaller effects at lower grade levels. Even so, Cooper (1989b) still recommended homework for elementary students because

homework for young children should help them develop good study habits, foster positive attitudes toward school, and communicate to students the idea that learning takes work at home as well as at school. (p. 90)

The Cooper, Robinson, and Patall (2006) meta-analysis found the same pattern of stronger relationships at the secondary level but also identified a number of studies at grades 2, 3, and 4 demonstrating positive effects for homework. In The Battle over Homework (2007), Cooper noted that homework should have different purposes at different grade levels:

For students in the earliest grades, it should foster positive attitudes, habits, and character traits; permit appropriate parent involvement; and reinforce learning of simple skills introduced in class.

For students in upper elementary grades, it should play a more direct role in fostering improved school achievement.

In 6th grade and beyond, it should play an important role in improving standardized test scores and grades.

Time Spent on Homework

One of the more contentious issues in the homework debate is the amount of time students should spend on homework. The Cooper synthesis (1989a) reported that for junior high school students, the benefits increased as time increased, up to 1 to 2 hours of homework a night, and then decreased. The Cooper, Robinson, and Patall (2006) study reported similar findings: 7 to 12 hours of homework per week produced the largest effect size for 12th grade students. The researchers suggested that for 12th graders the optimum amount of homework might lie between 1.5 and 2.5 hours per night, but they cautioned that no hard-and-fast rules are warranted. Still, researchers have offered various recommendations. For example, Good and Brophy (2003) cautioned that teachers must take care not to assign too much homework. They suggested that

homework must be realistic in length and difficulty given the students' abilities to work independently. Thus, 5 to 10 minutes per subject might be appropriate for 4th graders, whereas 30 to 60 minutes might be appropriate for college-bound high school students. (p. 394)

Cooper, Robinson, and Patall (2006) also issued a strong warning about too much homework:

Even for these oldest students, too much homework may diminish its effectiveness or even become counterproductive. (p 53)

Cooper (2007) suggested that research findings support the common "10-minute rule" (p. 92), which states that all daily homework assignments combined should take about as long to complete as 10 minutes multiplied by the student's grade level. He added that when required reading is included as a type of homework, the 10-minute rule might be increased to 15 minutes.

Focusing on the amount of time students spend on homework, however, may miss the point. A significant proportion of the research on homework indicates that the positive effects of homework relate to the amount of homework that the student completes rather than the amount of time spent on homework or the amount of homework actually assigned. Thus, simply assigning homework may not produce the desired effect-in fact, ill-structured homework might even have a negative effect on student achievement. Teachers must carefully plan and assign homework in a way that maximizes the potential for student success (see Research-Based Homework Guidelines).

Parent Involvement

Another question regarding homework is the extent to which schools should involve parents. Some studies have reported minimal positive effects or even negative effects for parental involvement. In addition, many parents report that they feel unprepared to help their children with homework and that their efforts to help frequently cause stress (see Balli, 1998; Corno, 1996; Hoover-Dempsey, Bassler, & Burow, 1995; Perkins & Milgram, 1996).

Epstein and colleagues conducted a series of studies to identify the conditions under which parental involvement enhances homework (Epstein, 2001; Epstein & Becker, 1982; Van Voorhis, 2003). They recommended interactive homework in which

Parents receive clear guidelines spelling out their role.

Teachers do not expect parents to act as experts regarding content or to attempt to teach the content.

Parents ask questions that help students clarify and summarize what they have learned.

Good and Brophy (2003) provided the following recommendations regarding parent involvement:

Especially useful for parent-child relations purposes are assignments calling for students to show or explain their written work or other products completed at school to their parents and get their reactions (Epstein, 2001; Epstein, Simon, & Salinas, 1997) or to interview their parents to develop information about parental experiences or opinions relating to topics studied in social studies (Alleman & Brophy, 1998). Such assignments cause students and their parents or other family members to become engaged in conversations that relate to the academic curriculum and thus extend the students' learning. Furthermore, because these are likely to be genuine conversations rather than more formally structured teaching/learning tasks, both parents and children are likely to experience them as enjoyable rather than threatening. (p. 395)

Going Beyond the Research

Although research has established the overall viability of homework as a tool to enhance student achievement, for the most part the research does not provide recommendations that are specific enough to help busy practitioners. This is the nature of research-it errs on the side of assuming that something does not work until substantial evidence establishes that it does. The research community takes a long time to formulate firm conclusions on the basis of research. Homework is a perfect example: Figure 1 includes synthesis studies that go back as far as 60 years, yet all that research translates to a handful of recommendations articulated at a very general level.

In addition, research in a specific area, such as homework, sometimes contradicts research in related areas. For example, Cooper (2007) recommended on the basis of 60-plus years of homework research that teachers should not comment on or grade every homework assignment. But practitioners might draw a different conclusion from the research on providing feedback to students, which has found that providing "feedback coupled with remediation" (Hattie, 1992) or feedback on "testlike events" in the form of explanations to students (Bangert-Drowns, Kulik, Kulik, & Morgan, 1991) positively affects achievement.

Riehl (2006) pointed out the similarity between education research and medical research. She commented,

When reported in the popular media, medical research often appears as a blunt instrument, able to obliterate skeptics or opponents by the force of its evidence and arguments. . . . Yet repeated visits to the medical journals themselves can leave a much different impression. The serious medical journals convey the sense that medical research is an ongoing conversation and quest, punctuated occasionally by important findings that can and should alter practice, but more often characterized by continuing investigations. These investigations, taken cumulatively, can inform the work of practitioners who are building their own local knowledge bases on medical care. (pp. 27-28)

If relying solely on research is problematic, what are busy practitioners to do? The answer is certainly not to wait until research "proves" that a practice is effective. Instead, educators should combine research-based generalizations, research from related areas, and their own professional judgment based on firsthand experience to develop specific practices and make adjustments as necessary. Like medical practitioners, education practitioners must develop their own "local knowledge base" on homework and all other aspects of teaching. Educators can develop the most effective practices by observing changes in the achievement of the students with whom they work every day.

Assign purposeful homework. Legitimate purposes for homework include introducing new content, practicing a skill or process that students can do independently but not fluently, elaborating on information that has been addressed in class to deepen students' knowledge, and providing opportunities for students to explore topics of their own interest.

Design homework to maximize the chances that students will complete it. For example, ensure that homework is at the appropriate level of difficulty. Students should be able to complete homework assignments independently with relatively high success rates, but they should still find the assignments challenging enough to be interesting.

Involve parents in appropriate ways (for example, as a sounding board to help students summarize what they learned from the homework) without requiring parents to act as teachers or to police students' homework completion.

Carefully monitor the amount of homework assigned so that it is appropriate to students' age levels and does not take too much time away from other home activities.

References

Balli, S. J. (1998). When mom and dad help: Student reflections on parent involvement with homework. Journal of Research and Development in Education, 31(3), 142-148.

Bangert-Drowns, R. L., Kulik, C. C., Kulik, J. A., & Morgan, M. (1991). The instructional effects of feedback in test-like events. Review of Educational Research, 61(2), 213-238.

Bennett, S., & Kalish, N. (2006). The case against homework: How homework is hurting our children and what we can do about it. New York: Crown.

Bloom, B. S. (1984). The search for methods of group instruction as effective as one-toone tutoring. Educational Leadership, 41(8), 4-18.

Cooper, H. (1989a). Homework.White Plains, NY: Longman.

Cooper, H. (1989b). Synthesis of research on homework. Educational Leadership, 47(3), 85-91.

Cooper, H. (2007). The battle over homework (3rd ed.). Thousand Oaks, CA: Corwin Press.

Cooper, H., Robinson, J. C., & Patall, E. A. (2006). Does homework improve academic achievement? A synthesis of research, 1987-2003. Review of Educational Research, 76(1), 1-62.

Corno, L. (1996). Homework is a complicated thing. Educational Researcher, 25(8), 27-30.

Epstein, J. (2001). School, family, and community partnerships: Preparing educators and improving schools. Boulder, CO: Westview.

Epstein, J. L., & Becker, H. J. (1982). Teachers' reported practices of parent involvement: Problems and possibilities. Elementary School Journal, 83, 103-113.

Fraser, B. J., Walberg, H. J., Welch, W. W., & Hattie, J. A. (1987). Synthesis of educational productivity research [Special issue]. International Journal of Educational Research, 11(2), 145-252.

Gill, B. P., & Schlossman, S. L. (2000). The lost cause of homework reform. American Journal of Education, 109, 27-62.

Good, T. L., & Brophy, J. E. (2003). Looking in classrooms (9th ed.). Boston: Allyn & Bacon.

Graue, M. E., Weinstein, T., & Walberg, H. J. (1983). School-based home instruction and learning: A quantitative synthesis. Journal of Educational Research, 76, 351-360.

Hattie, J. A. (1992). Measuring the effects of schooling. Australian Journal of Education, 36(1), 5-13.

Hoover-Dempsey, K. V., Bassler, O. C., & Burow, R. (1995). Parents' reported involvement in students' homework: Strategies and practices. The Elementary School Journal, 95(5), 435-450.

Kavale, K. A. (1988). Using meta-analyses to answer the question: What are the important influences on school learning? School Psychology Review, 17(4), 644-650.

Kohn, A. (2006a). The homework myth: Why our kids get too much of a bad thing. Cambridge, MA: Da Capo Press.

Kohn, A. (2006b). Abusing research: The study of homework and other examples. Phi Delta Kappan. 88(1), 9-22.

Kralovec, E., & Buell, J. (2000). The end of homework: How homework disrupts families, overburdens children, and limits learning. Boston: Beacon.

Marzano, R. J., & Pickering, D. J. (2007). Response to Kohn's allegations. Centennial, CO: Marzano & Associates. Available: http://marzanoandassociates.com/documents/KohnResponse.pdf

Marzano, R. J., & Pickering, D. J. (in press). Errors and allegations about research on homework. Phi Delta Kappan.

Marzano, R. J., Pickering, D. J., & Pollock, J. E. (2001). Classroom instruction that works: Research-based strategies for increasing student achievement. Alexandria, VA: ASCD.

National Education Commission on Time and Learning (1994). Prisoners of time. Washington, DC: U.S. Department of Education.

Paschal, R. A., Weinstein, T., & Walberg, H. J. (1984). The effects of homework on learning: A quantitative synthesis. Journal of Educational Research, 78, 97-104.

Perkins, P. G., & Milgram, R. B. (1996). Parental involvement in homework: A double-edge sword. International Journal of Adolescence and Youth, 6(3), 195-203.

Riehl, C. (2006). Feeling better: A comparison of medical research and education research. Educational Researcher, 35(5), 24-29.

Van Voorhis, F. (2003). Interactive homework in middle school: Effects on family involvement and science achievement. Journal of Educational Research, 96, 323-338.

Walberg, H. J. (1999). Productive teaching. In H. C. Waxman & H. J. Walberg (Eds.), New directions for teaching practice research (pp. 75-104). Berkeley, CA: McCutchen.

Wallis, C. (2006). Viewpoint: The myth about homework. Time, 168(10), 57.

Endnote

For a more detailed response to Kohn's views on homework, see Marzano & Pickering (2007) and Marzano & Pickering (in press).

Robert J. Marzano is a Senior Scholar at Mid-Continent Research for Education and Learning in Aurora, Colorado; an Associate Professor at Cardinal Stritch University in Milwaukee,Wisconsin; and President of Marzano & Associates consulting firm in Centennial, Colorado; robertjmarzano@aol.com. Debra J. Pickering is a private consultant and Director of Staff Development in Littleton Public Schools, Littleton, Colorado; djplearn@hotmail.com.

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The Case For and Against Homework - Educational Leadership

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March 2007 | Volume 64 | Number 6 Responding to Changing Demographics Pages 74-79 Robert J. Marzano and Debra J. Pickering Homework is typically defined as any tasks "assigned to students by school teachers that are meant to be carried out during nonschool hours" (Cooper, 1989a, p. 7).

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Through a series of exercises, the experimenters trained half the students to chalk up their errors to insufficient effort, and encouraged them to keep going. Those children learned to persist in the face of failure-and to succeed. The control group showed no improvement at all, continuing to fall apart quickly and to recover slowly.

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John Hattie developed a way of synthesizing various influences in different meta-analyses according to their effect size (Cohen's d). In his ground-breaking study "Visible Learning" he ranked 138 influences that are related to learning outcomes from very positive effects to very negative effects. Hattie found that the average effect size of all the interventions he studied was 0.40. Therefore he decided to judge the success of influences relative to this 'hinge point', in order to find an answer to the question "What works best in education?"
Originally, Hattie studied six areas that contribute to learning: the student, the home, the school, the curricula, the teacher, and teaching and learning approaches. (The updated list also includes the classroom.) But Hattie did not only provide a list of the relative effects of different influences on student achievement. He also tells the story underlying the data. He found that the key to making a difference was making teaching and learning visible. He further explained this story in his book " Visible learning for teachers ".

John Hattie updated his list of 138 effects to 150 effects in Visible Learning for Teachers (2011), and more recently to a list of 195 effects in The Applicability of Visible Learning to Higher Education (2015). His research is now based on nearly 1200 meta-analyses - up from the 800 when Visible Learning came out in 2009. According to Hattie the story underlying the data has hardly changed over time even though some effect sizes were updated and we have some new entries at the top, at the middle, and at the end of the list.

Below you can find an updated version of our first, second and third visualization of effect sizes related to student achievement.
Order VL for Teachers!

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Hattie effect size list - 256 Influences Related To Achievement

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Hattie's updated effect size list of 256 influences across all areas related to student achievement.

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Education Conferences, Professional Development For Teachers, Teacher Conferences

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Organizers of education conferences and professional development for teachers through continuing education teacher workshops. Specializing in neuroeducation and brain based teaching.

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Brain Based Learning | Brain Based Experts

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It is common knowledge that arts can arouse passions. Some believe that arts should be in school simply because many students thoroughly enjoy them. Others advocate a higher curriculum standing for arts on equal footing with math, science and language arts (Jensen). Now, there is now substantial evidence that arts are a "stand-alone" discipline. I would argue that arts support the neurobiological development of the brain in ways that enhance the social and academic performance of our students. This paper will explore some plausible mechanisms for how that happens.

This article introduces a new way to understand the impact of the arts on our students by asking three questions.
They are: Question One: What Really Drives Academic Success?
Question Two: Can the Underperforming Brain Be Changed?
Question Three: How Do Arts Change the Brain?
In this article, I'll asset that a very small group of sub-skills and attitudes drive academic success. I'll show that brains are highly adaptable and change everyday. Finally, I'll show that arts may alter the brain in positive ways that no other mediums can do. In sum, arts alter the neurobiological trajectory of the brain in ways that strengthen the academic and social skills unlike any other intervention.

The brain-based approach says, "How does this affect the brain?" The answer is that it's a powerful positive approach.

Introduction

For students to do well in school, their brain must function in ways that are academically and socially useful. We might loosely organize our students into three groups. There is a group that performs above the mean academically and they socialize reasonably well in a school environment.

Our second group brings special challenges. Students with severe AD/HD, head trauma, genetic disorders, fetal alcohol, oppositional disorders and Asperger's are just a sampling of the better-known special education challenges educators face today.

Finally, there is a much larger group of students that either underperform and/or misbehave which are given different labels such as unmotivated or "problem students." This article introduces a novel approach for understanding and succeeding with the last two groups of students. The assertion here is that it is possible to succeed with underperformers with fewer, more targeted interventions, in particular, the arts.

What do the arts bring to the table? The teachers are constantly trying new classroom strategies learned from books, trainings and conferences. The administrators are constantly inspiring, motivating and coaching their staff in endless ways to sharpen their collective "saw." Unfortunately, this approach of trying to get better performance from students and staff can become overwhelming. There seems to be no limit to the quantity of available strategies, so it becomes very much of a "hit or miss" approach. This results in a dizzying and endless stream of programs, themes, missions, projects and, ultimately, burnout among many educators.

But what if there was another way to go about this process. What if you could do less and get more? That's what arts do and we'll explore how that happens. The brain-based approach is a powerful positive approach.

Question One: What Really Drives Academic Success?

What is actually different in the brain that matters in the school context? I suggest the existence of multiple "operating systems" in the human brain, each of which actually determine success in school. These operating systems (e.g. academic, social, athletic, survival) contribute towards your student's success. But ultimately, since schools are all expected to reach performance goals, the academic operating system is of most relevance. Understanding this "system" is critical to a school's success.

The Academic "Operating System"

Collectively, we could dream up a list of the mind/brain attributes and the specific skills that students need to have to make it in school. But the list might get a bit cumbersome and, ultimately overwhelming for staff. The "operating system" model attempts to simplify this understanding. We can do that for a good reason. Our brain does not need to be perfect for a student to succeed in school. Academic success does not require perfection and that's good. School success requires "good enough" and that concept fits better with our brain. The aggregate of these factors is called our academic "operating system". Among the parts, we might include:

1. Effort: long-term motivation and the ability to defer gratification

2. Processing skills: auditory, visual and tactile

3. Attentional skills: engage, focus and disengage as needed

4. Memory capacity: short term and working memory

5. Sequencing skills: knowing the order of a process

6. Winner's mindset: "I can do it!" confidence

One does not need to be superior in all of these to get good grades. But one does need enough of each and any compensatory strategies to succeed. We all have a fault-tolerant brain, which is good at compensating and doing "work-arounds" so that one can succeed at a task even if it has to find an unconventional approach to get it done. The good news is that each of the critical processes in the brain's academic operating system (above) are malleable, trainable and can be improved. To make it more memorable, I have labeled the academic operating system "CHAMPS."

In this illustration, the six letters of the acronym refer to champion's mindset, hope, attention, memory, processing and sequencing.

Avoid dismissing this as simply another way to label study skills. While I am a strong supporter of learn to learn skills, this operating system approach describes the deeper neuropsychological underpinnings of academic success. After years of working with students teaching study skills, special education programs and mind/brain processes, what has emerged is a model featuring the fewest "moving parts" needed for school success. The attraction of this model is that it is simple enough for educators to understand, use and modify as needed. More importantly, it is based on a very powerful discovery in recent neuroscience: our brains have significant plasticity (Green CS, and Bavelier D., 2008).

See illustration below:

I assert that students who do well in school have a strong academic operating system. While it does not need to be perfect, there is a threshold that, when reached, ensures academic success. But before we explore it in more detail, is this discussion even relevant? Can the brain change? The brain-based approach says, "Of course that brain can change; it's designed to do that!"

Question Two: Can the Underperforming Brain Be Changed?

One of the most astonishing developments in cognitive science of the last hundred years is the debunking of the "fixed brain" myth. We now know that brain changes daily. The concept of brain plasticity reminds us that students are not stuck the way they are. While their success is dependent on their "operating systems", those systems can be upgraded.

We can increase mass in the student's brain (Dragons and May, 2008) and boost the student's production of new brain cells (Pereira, et al. 2007) and that's highly correlated with learning, mood and memory. We can add mass through effective vocabulary instruction (Lee, et al. 2007). When we teach thinking and processing skills, and it alters the brain (Levy (2007). Playing certain computer-aided instructional programs can increase attention and improve working memory, in just several weeks, (Kerns et al. 1999 and Klingberg, et al. 2005), both of which are significant "upgrades" to the student's operating system.

There are many things that effective schools do well. Whatever you've done in the past that worked for students, in some way, it has successfully changed the brain in ways that can be measured on achievement tests. Indirectly, we change the student's brains at school. When we teach students how to play an instrument, it changes brain mass (Gasser C, Shag G (2003). Many arts can improve attentional and cognitive skills (Gazzaniga, 2008). Playing chess can increase reading (Margulies, 1991) and math (Cage and Smith, 2000) vis a vis the pathways that increase attention, motivation, processing and sequencing skills.

The human brain is highly susceptible to environmental input. In some cases, change can happen, even permanent change, within minutes. But that's likely to be a change induced through trauma (emotional, psychological or physical). That change takes little time and can be quite lasting, but it's a negative change.

To get lasting positive change, you'll want skill-building because it is targeted towards the needs of the operating system. Each day, because learning consumes resources (e.g. glucose, time) there are upper limits on how much change per day. Go right up to the maximum allowable time per day by the brain. On average, that time is about 30-90 minutes of intensive skill building in any area. Beyond that, there's no evidence of gain. The brain just overloads and the change is dismissed. But a good arts program physically changes the brain. Good schooling has "upgraded" the student's operating system. In fact, every successful school intervention features a variation on the theme of "rebuild the operating system."

This system works on the principle of "fewest processes that matter most" to the learning process. But if you simply try to cram more content into the same brain, without "upgrading" the operating system, students will get bored, frustrated, and fail. This is where the arts come in.

Question Three: How Do Arts Change the Brain?

Five universities were funded as part of the Dan Foundation efforts to discover the effects of art on the human brain. Preliminary data after five years shows that arts positively impacts cognition (Posner, et al. 2008) Learning to play music is the best studied of the arts. New evidence suggests that music enhances cognition (Jonides, 2008). There is evidence from top neuroscientists in peer-reviewed journals such as The Journal of Neuroscience. The title, "Musical training shapes structural brain development" gives you a sense of the straightforward evidence (Hyde et. al. 2009). In fact, the brain-based approach says, all arts can affect the brain.

Here's the significance of a quality academic operating system: it has the capacity to override the adverse risk factors of injuries, toxins, divorce, poverty or a substandard childhood! When this operating system is "off" or simply not up to the constraints and demands of school, students struggle. These are not simple study skills; these are the capacity to focus, capture and discriminate information, process it, remember and represent it in a meaningful way. This bundled set of skills can be more than a great equalizer; it can be a critical piece of your school strategy for academic success with low- income students. When you address academic problems, it will always come back to strengthening the student's academic operating system. Stronger systems mean better performance.

Arts Build the Operating System

You can build the academic operating system with strategies that strengthen any of the subsystems that make up the academic operating system.

Champion's Mindset - This is the way of thinking that exudes confidence and willingness to try challenging tasks. Arts provide students ways to succeed as well as ways to showcase their skills. While it's tough for a language arts or science student to gain public acclaim, artists may develop positive mindsets through stage, concerts, productions, film and dance. Arts give students genuine affirmations often and support well-deserved student-to-student affirmations, too. Provide support for learning with tools, partners and confidence. Create short assignments and quick opportunities for quick successes that tell the learner, "You can do it!" Help strengthen their social status by providing appropriate opportunities for privileges with their peers. This attitude can be trained and it does change the brain (Duerden and Laverdure-Dupont 2008).

Hope - This is the feeling that long-term; it's worth "staying at it." It requires deferred gratification and only works when there is something to be hopeful for. Hope is the voice that says to you, "There are better days ahead." It is fundamental for long-term effort. Arts are not a "quick fix" and typically take months and years to develop significant strengths. This builds confidence and hope. Arts help strengthen teacher-to- student relationships so students know they have social support. Arts help set up situations where students can experience success. Arts provide quality role models or success. Arts teach imagination, positive goal setting and goal-getting. Arts help students learn to manage their time, create checklists to manage their lives. Arts teach how to make better choices and give practice in making choices. Arts invite dreams and let students draw, sing, talk, write or rap about them. Students with learned helplessness have a complete lack of hope, as do many with learning delays but not with arts. The skills of optimism are powerful and can be taught (Seligman 1998).

Attentional skills - Paying attention is not an innate skill. What's innate is shifting attention from one novel attention-grabber to another. It takes practice to learn to focus on the details over time. Arts training show that perception and sensory awareness can be enhanced (Ahissar, M. (2001) as can attentional skills (Hayes et al. 2003 and Eldar, et al. 2008). Focusing on high-interest content in arts where students can immerse themselves in a situation required detailed focus. Arts build focus through very high interest reading.

Arts build attention through focused practice in martial arts, dance, chess, model building and sports. Students with attention deficit typically have focus and attention issues (along with others), but arts can develop attentional skills. . Students are not stuck with poor attention span. Instead of demanding more attention in class, train students in how to build it.

Arts can make that happen; that's the brain-based approach.

Memory - In school, a strong memory is not just expected, it is priceless. We are all born with a good long-term memory for our "survival memories" such as where we live and work. We remember much of our spatial learning, emotional events, procedural and skill learning, conditioned response learning and highly behaviorally relevant data such as the names of our siblings, parents and cell number. Outside of those, school learning requires both short-term memory and long-term faculties. Music training strengthens verbal memory (Ho, et al. 2003). Dance strengthens sound and kinesthetic memory. Practice with simple call response in class. Build up to pair-share with partners. Strengthen memory with repetition, framing the importance of an idea. Develop their skills in mind mapping. This system can be enhanced through practice in many ways (Jaeggi et al.,2008).

Processing - This is the capacity to "flesh out" something. At the micro level, it means a student can process auditory input such as phonemes. That's quite important for reading. At the more macro level, it means the capacity to "process" an event such as being called a name, breaking up with a lover or forgetting to do your homework. We all need to know how to deal with difficulties, particularly emotional ones.

In addition, we need to be able to ask appropriate questions and think critically about a problem. We now know that the arts support math and science, two areas where processing is critical (Spelke, E. (2008). Arts provide the "how to" of processes because teachers either allow you to learn on your own or they walk you through steps. "Now we are doing this... and next we'll need to..." In other subjects, teachers often tell students what to do, but not how to do it. Arts can offer challenging games with structured practice time. Arts strengthen processing when students learn to play a musical instrument, design visual portfolios or take drama classes. Fortunately their brains can change when their system is targeted and treated. (Gaab, et al., 2007) and Erickson et al., 2007)

Sequencing - This is the set of skills that allows us to prioritize, identify and put in order a set of actions. When you cook and prepare a meal for guests, pack for a trip or paint a bedroom, you need sequencing skills. At school kids need sequencing for starting homework, writing a paper and planning a project. They need it for conflict resolution, doing a math problem or planning their day. We now know that arts support reading skills (Wandell, B., et al. 2008). In addition to dance, music playing can also strengthen both tactile and auditory memory (Ragert et al. 2004). All music requires and develops sequencing skills. Even the visual artist has to sequence the thinking, the shapes, proportions color and contrast. Visual arts give students the opportunity for building things (assembling models, paper projects or project displays). Be the guide for their time invested in organizing their work, writing a paper or problem solving. Most art opportunities build and strengthen attention, hope, processing, and sequencing. This is a skill set than can change through practice.

How to Maximize Results

Building the operating system is a top priority. If a student is in an impoverished environment for hours at home a day, then gets a few minutes of operating system enhancement at school, that time constitutes only a tiny fragment of the student's total week. That time is unlikely to produce substantial long-term benefits, though it's certainly better than nothing. The message her is clear: do not kid yourself about changing the brain: it takes focused effort over time. To get lasting positive change, you'll want to go right up to the maximum allowable time per day by the brain. On average, that time is about 30-90 minutes of arts will build a student's operating system. Beyond that, there's no evidence of gain. The brain just overloads and the change is dismissed.

Let's translate this into your school time. To maximize change with the arts in mind, it's got to be a priority. Unless one follows the "brain's rules" for skill-building, precious time is wasted, students learn less and teachers get frustrated. The rules are simple. The learner must buy into the activity, perceive relevance, pay focused attention and get enough rest at night. The actual activity must be coherent, have built-in positive and negative feedback, last at least 30-90 minutes a day as well as for 3-6 days a week. If you don't follow the rules, your results will be delayed. Arts are one of the best options for building the operating system. When we lose out on arts, we lose out on more than student enjoyment or potential career options. We lose out on the critical brain development for immediate academic success and lifelong learning.

Summary

The brain-based approach says, "How does music affect the brain?" The answer is that it's a powerful positive approach.

We explored three questions.

The first question asked us, "What Really Drives Academic Success?" This question was addressed by suggesting that six factors comprise an "academic operating system" and that they matter more than anything else within our brain. While we have other, parallel "operating systems, the academic one is a big piece of the achievement score solution.

Question two was, "Can the Underperforming Brain Be Changed?" Yes, it can! The scientific evidence shows that it can be changed. Of particular interest is that studies show the academic operating system can be strengthened with targeted skill-building.

Our third question was, "How Do Arts Change the Brain?" By using the operating system as a vehicle for progress, arts give students a practical path for success. Only a few things really matter. Those few things can be improved with targeted practice. Arts are a formidable resource for building the student's operating system. But now, we have to put it in action. Support the arts for any reason you want. But arts can and do hold their own when it comes to improving the lives of our students.
All aboard? Eric Jensen is a former teacher at all levels, including three universities. He co-founded Jensen Learning Corporation, an international professional training organization that synthesizes brain research and its applications for educators (www.jensenlearning.com). Jensen is a member of the International Society for Neuroscience, New York Academy of Science and is has a PhD in Human Development. Jensen has authored over 25 books including Arts with the Brain in Mind, Teaching with the Brain in Mind, Different Brains, Different Learners and Enriching the Brain. He can be reached at info@jensenlearning.com.
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How Arts Change the Brain - Brain Based Learning | Brain Based Experts

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It is common knowledge that arts can arouse passions. Some believe that arts should be in school simply because many students thoroughly enjoy them. Others advocate a higher curriculum standing for arts on equal footing with math, science and language arts (Jensen). Now, there is now substantial evidence that arts are a "stand-alone" discipline.