Saturday, October 1, 2011

Integrating psychology and mathematics via art: the "Impressionism Project".

Integrating psychology and mathematics via art: the "Impressionism Project". STUDENTS LEARN BETTER when they relate their lessons to their ownexperiences. According to according toprep.1. As stated or indicated by; on the authority of: according to historians.2. In keeping with: according to instructions.3. the Ohio Department of Education'srecently adopted Academic Content Standards for Social Studies: Learning is enhanced when students make meaningful connections between new information that they are learning and their own experiences. Combining social studies instruction with the study of other disciplines such as art and literature helps to reinforce the learning within each discipline. It also helps the students to develop conceptual frameworks that lead to broader understandings. (State Board of Education of Ohio, 2002, p.2) After eighteen years in the classroom, I agree wholeheartedly whole��heart��ed?adj.Marked by unconditional commitment, unstinting devotion, or unreserved enthusiasm: wholehearted approval.whole withthis assessment. As educators in Ohio, we must provide students with the tools andopportunities that enable them to excel in ways specified by the OhioDepartment of Education. Students gain a more thorough understanding andretain more information when they study material that they believe isrelevant, and when they "learn by doing." This is especiallytrue in social studies classes. Unfortunately, many teachers stillbelieve that they cannot educate pupils this way in a typical socialstudies classroom. It is time to debunk de��bunk?tr.v. de��bunked, de��bunk��ing, de��bunksTo expose or ridicule the falseness, sham, or exaggerated claims of: debunk a supposed miracle drug. this myth. Meaningful connections can be madein diverse areas, but only if educators are willing to pursue uncommonapproaches to teaching and learning. Traditionally, math and social studies are rare partners in theclassroom. Yet, when looking at the Ohio Department of Education'sAcademic Content Standards for Mathematics, one finds that: Making connections ... between mathematics and other disciplines is critical for student success in using mathematics effectively ... Students in the classroom need opportunities to recognize and draw upon the connections between and among topics studied. Teachers can facilitate this process by engaging students in multi-layered problem solving situations that cross disciplines and engage students' interests." (State Board of Education of Ohio, 2001, p.198) This excerpt ex��cerpt?n.A passage or segment taken from a longer work, such as a literary or musical composition, a document, or a film.tr.v. ex��cerpt��ed, ex��cerpt��ing, ex��cerpts1. about math studies closely echoes the above socialstudies excerpt, and for good reason. There are many ways to "grab" students' attention,maintain their interest, and engage them in unique and challengingexperiences. One strategy I have found successful involves the use ofcross-curricular approaches. Since social studies encompass a diversearray of subject areas, myriad connections can be made between thisfield and others. Discovering those connections is what makes learningexciting for students and teaching exhilarating ex��hil��a��rat��ing?adj.Causing exhilaration; invigorating.ex��hila��rat for educators. In thisarticle, I will share a recently designed and implemented project thatsuccessfully integrates social studies and mathematics by utilizing thevehicle of visual arts visual artsnpl → artes fpl pl��sticasvisual artsnpl → arts mpl plastiquesvisual artsnpl → . The Problem with the Standard Approach In my senior psychology class, I have been impressed im��press?1?tr.v. im��pressed, im��press��ing, im��press��es1. To affect strongly, often favorably: by howinterested students become when we study sensation and perception. Theyseem particularly enthralled en��thrall?tr.v. en��thralled, en��thrall��ing, en��thralls1. To hold spellbound; captivate: The magic show enthralled the audience.2. To enslave. by how our minds can be "tricked"into seeing or believing certain things. To demonstrate this effect inconcrete terms, I show movies such as The Sixth Sense, The Others, orThe Ring. These movies set viewers up and grab them at the end with"Aha!" moments, similar to techniques utilized in the oldTwilight Zone twilight zone - [IRC] Notionally, the area of cyberspace where IRC operators live. An op is said to have a "connection to the twilight zone". television series. These visual examples prove to beeffective tools in relating student experiences to perception andsensation. The climactic cli��mac��tic? also cli��mac��ti��caladj.Relating to or constituting a climax.cli��macti��cal��ly adv.Adj. 1. "twist" in these movies succeeds in"fooling" them and they are genuinely intrigued by the factthat they were "duped." However, I wanted students to go a step further, and I challengedthe class to reject the notion that such "fooling and duping Duping refers to the practice of exploiting a bug in a video game to illegitimately create duplicates of unique items or currency in a persistent online game, such as an MMOG. "is a natural occurrence. I did not want them to experience this feelingof awe and then let it slip away without further analysis. This hashappened in the past. I determined to seize upon this teachable teach��a��ble?adj.1. That can be taught: teachable skills.2. Able and willing to learn: teachable youngsters. momentand lead my students to learn why and how this phenomenon occurs. Inother words Adv. 1. in other words - otherwise stated; "in other words, we are broke"put differently , I wanted the students to examine different techniques usedto achieve these effects; to think about how they think and analyze whytheir brains and senses operate the way they do; to move beyondsensations, and ascertain causes; to delve deeply into the realm oftheir own perceptions. Through such personal analysis, the studentswould look more deeply into these occurrences and their responses."Learning by doing" would be conducive con��du��cive?adj.Tending to cause or bring about; contributive: working conditions not conducive to productivity.See Synonyms at favorable. to accomplishing thistask. I decided to use a different form of art to demonstrate how ourbrains perceive in relation to our senses. I introduced students to thework of artist M.C. Escher, who uses a variety of methods to visuallydeceive TO DECEIVE. To induce another either by words or actions, to take that for true which is not so. Wolff, Inst. Nat. Sec. 356. viewers. In some sketches, Escher creates multiple illusions ofendless staircases leading nowhere. He also depicts a waterfall waterfall,a sudden unsupported drop in a stream. It is formed when the stream course is interrupted as when a stream passes over a layer of harder rock—often igneous—to an area of softer and therefore more easily eroded rock; the edge of a cliff or feedinginto a canal that is, amazingly, when followed by the eye, the source ofthe waterfall that feeds it. In other pictures, he explorestessellations, which appear to go on ad infinitum ad in��fi��ni��tum?adv. & adj.To infinity; having no end.[Latin ad, to + . Woven throughout hiswork are interesting creatures, some of which exist, some which do not.Yet each illusion works. (Incidentally, excellent Escher reproductionsmay be found in books and on the Internet, and images from commerciallyavailable calendars may also be passed around among students.) In the past, I taught this unit by displaying and discussing theseworks of art in the light of relevant psychological principles. Afterthe initial wonderment, students discussed in great detail how these"deceptions" were achieved. I would ask students to writeabout their perceptions of a drawing that appealed to them; then I wouldshare Escher's original intentions in creating that piece of art.However, I was still frustrated frus��trate?tr.v. frus��trat��ed, frus��trat��ing, frus��trates1. a. To prevent from accomplishing a purpose or fulfilling a desire; thwart: because, after looking, discussing, andwriting, the students generally forgot what they had experienced. Iwanted to find a better way to engage students' minds in this realmof sensation and perception, one they would not so easily forget. Uncommon Approach Escher once commented, By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I had made, I ended up in the domain of mathematics. Although I am absolutely innocent of training or knowledge in the exact sciences, I often seem to have more in common with mathematicians than with my fellow artists. (Escher, 2001, p.6.) In fact, Escher was being quite modest. Due to the high level ofmathematical properties inherent in his varied and unique designs, manyof his creations were acclaimed ac��claim?v. ac��claimed, ac��claim��ing, ac��claimsv.tr.1. To praise enthusiastically and often publicly; applaud. See Synonyms at praise.2. as masterpieces Masterpieces was released in Japan and Australia in anticipation for Bob Dylan's 1978 tour. It is more or less a greatest hits collection spanning his entire career up that point. by mathematicians Mathematicians by letter: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z See alsoRequested mathematicians articles (by country, etc.) List of physicists External links . Fromtessellations to Mobius strips to impossible structures, Eschereffectively incorporated math into his designs in highly unusual ways. Intrigued by the mathematical concepts in Escher's work, Iwanted to convey to students these aspects, as well as the psychologicalperspectives. However, like Escher, I felt I lacked the training to doso. Fortunately, I have access to the expertise of a superb mathteacher, Dale Price, who also has a fascination for Escher. Dale is aunique educator, certified See certification. in history and holding a master's degree master's degreen.An academic degree conferred by a college or university upon those who complete at least one year of prescribed study beyond the bachelor's degree.Noun 1. in mass communication. He also is a published author, having co-writtena high school journalism text. Dale and I find Escher's work fascinating for differentreasons. Not surprisingly, we approach his work from the perspective ofour academic disciplines. As we discussed possibilities, we immediatelyrecognized numerous connections and realized that our students may belooking at Escher's artwork in different ways, too. As a result, wedevised a lesson plan, a joint effort between our areas of interest.Without a doubt, the resulting Impressionism impressionism, in paintingimpressionism,in painting, late-19th-century French school that was generally characterized by the attempt to depict transitory visual impressions, often painted directly from nature, and by the use of pure, broken color to Project enhanced our growthas much as it did that of our students. Incidentally,"impressionism," as Dale and I used the term in theproject's early developmental stages, refers to the practice ofpresenting and elaborating on one's reactions, or impressions, to awork of art; it is not related to the art movement from the latenineteenth and early twentieth centuries. Preliminary Steps Psychology is the scientific study of human behavior and mentalprocesses, in other words, what people think, feel, and do. (Kasschau,2003.) After I had explained this term to my students, I informed themthat this was exactly what we were about to do. First, in psychology class, we defined the concepts of sensationand perception. Sensation is an awareness or mental process due tostimulation of a sensory sensory/sen��so��ry/ (sen��sor-e) pertaining to sensation. sen��so��ryadj.1. Of or relating to the senses or sensation.2. organ: what occurs when a stimulus activates areceptor. (Kasschau, 2003.) For example, we have a sensation when oureyes are stimulated by a strange greenish tinge in the color of the sky.Perception refers to insight and our capacity for comprehension comprehensionAct of or capacity for grasping with the intellect. The term is most often used in connection with tests of reading skills and language abilities, though other abilities (e.g., mathematical reasoning) may also be examined. : theorganization of sensory information into meaningful experiences.(Kasschau, 2003.) Organizing the sensation of that strange green hue inthe sky when a tornado tornado,dark, funnel-shaped cloud containing violently rotating air that develops below a heavy cumulonimbus cloud mass and extends toward the earth. The funnel twists about, rises and falls, and where it reaches the earth causes great destruction. touched down into something we understand, forexample, is a perception. Sensation and perception enable us to gatherand interpret information in our environments. The interesting twist isthat one or both can be fooled; how this occurs is what I wanted mystudents to examine further. Second, in math class, we scrutinized the related mathematicalterms to clarify meaning for students. Tessellations have a checkered check��ered?adj.1. Divided into squares.2. Marked by light and dark patches; diversified in color.3. Marked by great changes or shifts in fortune: a checkered career. appearance and are similar to mosaics. Mosaics extend over given areaswithout leaving any region uncovered, so the geometric meaning (mathematics) geometric mean - The Nth root of the product of N numbers.If each number in a list of numbers was replaced with their geometric mean, then multiplying them all together would still give the same result. of theword "tessellate tes��sel��late?tr.v. tes��sel��lat��ed, tes��sel��lat��ing, tes��sel��latesTo form into a mosaic pattern, as by using small squares of stone or glass. " is to cover the plane with a pattern thatleaves no region uncovered. (Schwartzman, 1994.) By extension, space orhyperspace hyperspace - /hi:'per-spays/ A memory location that is *far* away from where the program counter should be pointing, often inaccessible because it is not even mapped in. (Compare jump off into never-never land. may also be tessellated tessellated/tes��sel��lat��ed/ (tes��ah-lat?ed) divided into squares, like a checker board. tes��sel��lat��edadj.Composed of or patterned in small squares. . A Mobius strip is a continuousone-sided surface representing infinity infinity,in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. . (Schwartzman, 1994.)"Impossible" figures or structures can be drawn in twodimensions, but cannot be constructed in three. We reviewed otherrelated mathematical terminologies, providing students with informationabout geometric designs, perspective, symmetry symmetry,generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences. , reflection, congruency con��gru��en��cy?n. pl. con��gru��en��ciesCongruence. ,infinity, and other applicable principles. Third, we assembled the following list of resources we believedstudents would find useful during the introductory phase of thisproject: 1. M.C. Escher: The Graphic Work (an interpretive in��ter��pre��tive? also in��ter��pre��ta��tiveadj.Relating to or marked by interpretation; explanatory.in��terpre��tive��ly adv. book by theartist). 2. M.C. Escher (a book by Sandra Forty). 3. The Fantastic World Of M.C. Escher (a video about his work andlife). 4. http://www.mcescher.com/ (the official Escher website). Through resources such as these, our students examinedEscher's techniques in relation to each subject area. Educators cansupplement these resources as needed as neededprn. See prn order. . For our project, however, this wasjust the preliminary phase; the real learning experience was about tobegin. A "Hands-on Minds-on" Project Anything which can be called a study, whether arithmetic, history, geography, or one of the natural sciences, must be derived from materials which at the outset fall within the scope of ordinary life experience. (Dewey, 1938, p.73.) The Impressionism Project was originally intended to draw outrelationships between mathematics and psychology through the medium ofvisual arts. In reality, the successful implementation of this projectincorporated many subject areas and student personal experiences. Thepossibilities seem as endless as the ideas in the artistic expressionsthe students presented. Results are guaranteed to be unique, fresh, and"different." However, we expected the students to strongly emphasize bothmathematical principles and psychological concepts in their finisheddesigns. We would not judge artistic ability, only the application ofconcepts and principles in relation to the artwork. Two sets ofchallenges were provided to students, who then had approximately twoweeks to complete each list of instructions. Of course, time is aprecious and negotiable NEGOTIABLE. That which is capable of being transferred by assignment; a thing, the title to which may be transferred by a sale and indorsement or delivery. 2. commodity, and educators can decide just howmuch in-class time they want to commit to this worthwhile endeavor. Aswith many similar projects, students can complete a great deal of thiswork outside of the classroom. The first challenge described styles the students could emulate em��u��late?tr.v. em��u��lat��ed, em��u��lat��ing, em��u��lates1. To strive to equal or excel, especially through imitation: an older pupil whose accomplishments and style I emulated.2. : * Draft a tessellation/tiling, by hand, using a Mira or computersoftware. * Incorporate a Mobius strip (representing infinity) into adrawing. * Create an "impossible figure," an optical illusion thatcan exist only on paper. * Draw a stripe stripe - data striping image. * Create a surrealistic sur��re��al��is��tic?adj.1. Of or relating to surrealism.2. Having an oddly dreamlike or unreal quality.sur��re picture: focus on "dualisms" ordouble images. * Draw an "upside-down" image that reveals two distinctimages depending on which way picture is viewed. * Make a working kaleidoscope kaleidoscope(kəlī`dəskōp), optical instrument that uses mirrors to produce changing symmetrical patterns. Invented by the Scottish physicist Sir David Brewster in 1816, the device is usually a hand-held tube, a few inches to as much . * Create a "combination effect" of your choosing. Students could produce one or several drawings, but the challengeemphasized the quality of the result more than the quantity. Thus, oneartistic work per pupil was the norm. The second challenge required students to perform a self-analysis,to really think about how and why they had designed their creations theway they did. They were required to explain or describe the following: * Where their idea had come from; how they had been influenced. * How the piece was created. * The illusion/effect produced and how it was successfullyachieved. * How positive and negative space contributed to the overalleffect. * Whether the image in the artwork could be physically constructed.(Extra points for trying it!) * Discoveries they made as they implemented their design. * Problems encountered and solved as the creation evolved. * Mathematically, what ideas they incorporated into their designsand what role those ideas played as the piece was created. * Psychologically, how the artistic creation works, how it can beexplained through psychological concepts and/or philosophies. During the second challenge, students were engaged in a process ofself-analysis, focusing on how and why they made each decision abouttheir artwork. This definitely contrasts with simply viewing someoneelse's artwork, discussing, and making assumptions about theartist's intent and style, and then moving on to the next piece. Inthis phase, students have become the artists, the experts who can andmust provide answers to the questions asked. As a result, pupil interestlevels remain high as curricular connections are firmly establishedbetween the "unusual" combination of art, math, andpsychology. Throughout the whole project, the lessons enhance thestudents' critical thinking skills as they simultaneously thoughtabout the art and analyzed an��a��lyze?tr.v. an��a��lyzed, an��a��lyz��ing, an��a��lyz��es1. To examine methodically by separating into parts and studying their interrelations.2. Chemistry To make a chemical analysis of.3. their thinking. This reflective sectiondistinguishes the Impressionism Project and invites objective assessmentby educators outside the field of art. Learning Objectives To engage students in unique thinking and interpretation processes,we took the following learning objectives as goals: * To foster an appreciation amongst students for the myriad wayspsychology and mathematics can enrich their lives through artwork. * To enable students, through observation, analysis, andevaluation, to better understand the relationships between these areasof study, so that they can watch for other connections when theyencounter similar situations in the future. * To encourage students to "think outside the box," asthey discover and enhance their cognitive and hands-on skills. * To reveal the inherent beauty associated with mathematicalprinciples, abstract as well as concrete in nature; to cast a new lighton math by dispelling stereotypical notions about the subject. * To assist students to organize their sensations and perceptionsinto meaningful experiences; to demonstrate how these human qualitieswork together to help them interpret and understand reality. Lessons Learned Only minor problems were encountered as the project unfolded. Inthe early phase, we needed to address performance fears revealed byremarks like, "I'm not an artist," "I can'tdraw," and "I don't know Don't know (DK, DKed)"Don't know the trade." A Street expression used whenever one party lacks knowledge of a trade or receives conflicting instructions from the other party. where to begin." Somestudents were apprehensive about doing math in psychology class andvice-versa. However, as every educator knows, these kinds of reactionsare common when students hear the phrase "cross-curricularproject." In a turn-about move, I initiated a discussion, in whichstudents participated enthusiastically, about perceptions and theinfluence they can have, for better or worse, in our lives, and studentfears and negativity diminished. More importantly, we established afoundation for the Impressionism Project in a collaborative fashion. We did uncover a problem in the second challenge that caused ussome concern. Some of the self-analysis reports were not well-balancedbetween issues of math and issues of psychology. In other words, thesewritten reflective pieces focused far too heavily on one subject andslighted the other. This unexpected problem led to assessmentdifficulties and made it hard to judge each project in a consistentmanner. To avoid this problem in the future, we intend to provide morespecific expectations in the instructions for that challenge. Also,since the Impressionism Project has already been implemented, we willuse former student examples to ensure better understanding amongst laterparticipants. The Ultimate Assessment Tool As mentioned earlier, artistic ability was not evaluated.Assessment focused on how students justified the relationship of theirartwork to the psychological principles of sensation and perception, inconjunction with the successful use of mathematical principles. Art wasthe vehicle used to arrive at this destination. Since assessment is amatter of concern for educators, students have the opportunity tocomment on their overall experiences. I regard student feedback as theultimate assessment tool. With their permission, I have included somestudent responses to the Impressionism Project below. [ILLUSTRATION OMITTED] "We were given so much freedom to express ourselves, I found myself working on this project at home for hours. My idea came from M.C. Escher's drawing Waterfall. I took the idea of impossible figures and expanded on it by making a system of more complex waterways. Escher inspired my drawing, so it has a similar philosophical idea. I wanted to use what people see as normal objects, but then twist them by turning them into an impossible figure. This causes viewers to take more than just one look and it leaves them wondering what it would be like to be there and what the structure could be used for. Also, viewers are intrigued and surprised when they realize that they can never be in a similar place." --MARK DOMSCHOT "This project helped me explore artistic talents that I never knew were there." --LAJUANDA MCINTYRE "It wasn't until I tried to think about how I think that I actually discovered how hard of a job this project was going to be. This is called an Impressionism Project, but in reality human nature as a whole is one big Impressionism Project! The influence for my drawing came from my life; all of my life, everything that has ever happened to me plays a role in what this project is." --SHANNON GLADIEUX [ILLUSTRATION OMITTED] "Creating my own representation of a three-dimensional image on a two-dimensional surface was hard work, but a fascinating and fulfilling experience." --JAMES HOUGHTELL "This senior Impressionism Project allowed me to better understand math, psychology and, ultimately, myself." --JOSH RUTT [ILLUSTRATION OMITTED] "As I began to draw I saw nothing at first, but as my work progressed I began to relate the project to myself. It symbolizes all that my life is centered around. The mathematical ideas in my design mainly focus around geometric figures. There are squares and triangles that help to make odd shapes and cause a person's eyes to wander around the piece rather than remain in one small focused area. They also encourage people to rotate the drawing and try to see as many figures as possible. The effect of this drawing is to force people to truly think about what it could be without being told what it is. My creation works by drawing people's eyes across the paper moving from positive to negative space. This encourages the viewer to see different designs as their eyes pass from one form and back to it. This is true for both perception and sensation because what the person sees definitely gives them a sensation." --SARAH TROUTMAN "The senior Impressionism Project gave me many creative ideas. These ideas were non-stop; they kept on coming and expanded into other ideas. During the project I could not stop thinking about it at all. Now I'm looking forward to working on many other artistic ideas on my own time." --STEVEN BARKER "My drawing is an image of how I think. Our psychology and math Impressionism Project opened the window to my imagination and talent. I was able to express myself on my own creative level. I never knew the height of my imagination and creativity until participating in this art project." --JOHN MATTONI "I loved discovering the relationship between psychology, math, and art." --JENNIFER KUSZ Discovering Wonder Based on positive responses such as these, I have come to theconclusion that the Impressionism Project is a keeper. For two weeks,students were immersed im��merse?tr.v. im��mersed, im��mers��ing, im��mers��es1. To cover completely in a liquid; submerge.2. To baptize by submerging in water.3. in an atmosphere of discovery, and they becameintimately acquainted with concepts perhaps never contemplated before.Students were moved to a high degree of introspection introspection/in��tro��spec��tion/ (in?trah-spek��shun) contemplation or observation of one's own thoughts and feelings; self-analysis.introspec��tive in��tro��spec��tionn. and theydemonstrated a firm understanding of higher-order thinking Higher-order thinking is a fundamental concept of Education reform based on Bloom's Taxonomy. Rather than simply teaching recall of facts, students will be taught reasoning and processes, and be better lifelong learners. skills. By taking the next step and allowing them to design their ownartistic creations, we enabled our students to experience art in aunique way, made possible by the rare marriage of psychology andmathematics. Pupils saw the connections they were literally"making," as they fashioned their own works of art, and thenjustified these artistic creations in psychological and mathematicalterminology. Once they became the artists, students were transformed frompassive viewers into active participants in the learning and teachingprocess. To some extent, novices became experts, sharing what and howthey learned as they learned, consequently educating those around them.For these reasons, I am certain that these students will remember thisexperience, because it held meaning for each participant on anindividual level, yet benefited everyone involved. Regarding his artistic creations, M.C. Escher once said, The ideas that are basic to them often bear witness to my amazement and wonder at the laws of nature which operate in the world around us. He who wonders discovers that this is in itself a wonder." (Escher, 2001, p.6.) In relation to the Impressionism Project, all participants seemedto enjoy each phase of the activity, even those with preconceivednotions Noun 1. preconceived notion - an opinion formed beforehand without adequate evidence; "he did not even try to confirm his preconceptions"parti pris, preconceived idea, preconceived opinion, preconception, prepossession about art, math or psychology. Educators as well as studentsprofited from this unique experience that truly led us with thediscovery of wonder. REFERENCES Dewey, J. (1938). Experience & Education. New York New York, state, United StatesNew York,Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of : MacmillanPublishing Company. Escher, M.C. (2001). M.C. Escher: The Graphic Work. Germany: Barnes& Noble, Inc. Kasschau, R. (2003). Understanding Psychology. Columbus, Ohio Columbus is the capital and the largest city of the American state of Ohio. Named for explorer Christopher Columbus, the city was founded in 1812 at the confluence of the Scioto and Olentangy rivers, and assumed the functions of state capital in 1816. :Glencoe/McGraw-Hill Ohio Department of Education (2001). Academic Content Standards forMathematics. Columbus, Ohio. Ohio Department of Education (2002). Academic Content Standards forSocial Studies. Columbus, Ohio. Schwartzman, S. (1994). The Words of Mathematics. Washington DC:Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on undergraduate mathematics education. Members include teachers at the college and high school level; graduate and undergraduate students; and mathematicians and scientists. . JOSEPH HUTCHINSON* * Joseph Hutchinson has taught social studies at Toledo TechnologyAcademy The neutrality and factual accuracy of this article are disputed.Please see the relevant discussion on the . for the past 6 years. He is co-director of the Toledo AreaWriting Project, and a member of the Ohio Council for the Social Studiesand the National Council of Teachers of English MissionAs stated on their official website, the NCTE ( National Council of Teachers of English) is a professional organization dedicated to "improving the teaching and learning of English and the language arts at all levels of education. .

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