Social constructivism in practice: case study of an elementary school's mathematics program.

Social constructivism in practice: case study of an elementary school's mathematics program. AbstractThis research investigated implications for the implementation ofsocial constructivist epistemology Constructivism is a perspective in philosophy that views all of our knowledge as "constructed", under the assumption that it does not necessarily reflect any external "transcendent" realities; it is contingent on convention, human perception, and social experience. on teaching/learning of mathematicsin a K-4 public school with particular focus on African American African AmericanMulticulture A person having origins in any of the black racial groups of Africa.See Race. fourthgrade students. In addition, the study examined the impact of socialconstructivist con��struc��tiv��ism?n.A movement in modern art originating in Moscow in 1920 and characterized by the use of industrial materials such as glass, sheet metal, and plastic to create nonrepresentational, often geometric objects. theory on the structure and culture of the school.Constructivist inquiry was used to make sense of the data. In thispaper, there is a discussion of lessons learned from this study withparticular emphasis on structural changes, cultural changes, politics ofreforming mathematics education, and the impact of social constructivistteaching on African American students' achievement.**********There is much research about how students learn mathematics and howmathematics ought to be presented to young children (Burns, 1992;Campbell Campbell, city, United StatesCampbell,city (1990 pop. 36,048), Santa Clara co., W Calif., in the fertile Santa Clara valley; founded 1885, inc. 1952. , 1996; Cobb, Wood, & Yackel, 1991; Cobb & Yackel, 1996;Fennema, Franke Franke is a Swiss company involved primarily in the production of stainless steel and composite plastic sinks and taps. It is also involved in the making of kitchen systems such as cookers, kitchen accessories such as strainer bowls and food preparation platters. , Carpenter, & Carey See also: CaryCarey is the name of several places: United Kingdom Carey, Herefordshire Carey, Northern Ireland United States Carey, Alabama Carey, Georgia Carey, Idaho , 1993; Romberg Rom��berg? , Sigmund 1887-1951.Hungarian-born American composer of operettas, including Blossom Time (1921) and The Student Prince (1924).Noun 1. , Shafer, &Webb, 2000; Simon, 1995; Wheatley & Reynolds, 1999; Yackel, 1995).In addition, research documents indicate that mathematics instructiondoes not provide students with opportunities to acquire deepmathematical understanding (National Council of Teachers of Mathematics The National Council of Teachers of Mathematics (NCTM) was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally. [NCTM NCTM National Council of Teachers of MathematicsNCTM Nationally Certified Teacher of MusicNCTM North Carolina Transportation MuseumNCTM National Capital Trolley MuseumNCTM Nationally Certified in Therapeutic Massage ], 2000a).A National Science Foundation (NSF NSF - National Science Foundation , 1996) report indicates fourthgrade students in the United States United States,officially United States of America, republic (2005 est. pop. 295,734,000), 3,539,227 sq mi (9,166,598 sq km), North America. The United States is the world's third largest country in population and the fourth largest country in area. performed better on mathematicsproficiency tests See aptitude tests. of basic skills compared to previous test results.However, the report exposes students' lack of conceptualunderstanding of those basic skills. Furthermore, current research showsthat minority students (i.e., Native Americans, Hispanics, and AfricanAmericans) perform well below the national average (Ladson-Billings,2001; NCTM, 2000b, 2000c; 2001a, 2001b, 2001c).This study communicates the complexity of reforming mathematicseducation with particular focus on instruction for underachievingAfrican American students. Moses & Cobb (2001) posit that if allstudents can learn mathematics, then they ought to be provided withopportunities to learn mathematics. Therefore, mathematics literacy is aright rather than a privilege for a few. This study may offer some ideasabout how to raise mathematics achievement among African Americanstudents who have scored low compared to the average national scores onmathematics achievement tests.This project investigated the following questions: (1) Can socialconstructivist epistemology be implemented to raise achievement levelsof African American students? (2) What impact does social constructivisttheory have on the structure and culture of school? And (3) What are thesocial and political implications for reforming mathematics education ina K-4 elementary school elementary school:see school. ? The research tells the story of a K-4elementary school that struggled to reform mathematics education(1990-2003) by implementing social constructivist theory and pedagogy inclassrooms. After eight years (1990-1998) of focusing on restructuring restructuring - The transformation from one representation form to another at the same relative abstraction level, while preserving the subject system's external behaviour (functionality and semantics). and recapturing mathematics classrooms, fourth grade students'scores on state mathematics tests dramatically improved for a five yearperiod: 1999-2003. The test results attracted local and statewideattention because African American and white students achieved at aboutthe same high level. In what follows, we describe some history of thereforms in this K-4 school. Then, we discuss our theoretical framework,the design of the study, and lessons learned.A Bit of HistoryThe school enrolls 525 students: 60% African American, 34% white,and 6% other racial/ethnic groups. This school is one of five K-4elementary schools in a Midwest Midwestor Middle West,region of the United States centered on the western Great Lakes and the upper-middle Mississippi valley. It is a somewhat imprecise term that has been applied to the northern section of the land between the Appalachians school district that is racially andeconomically diverse. Eighty-five percent of the teaching staff holds aMaster'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. . The principal, former assistant principal, and twoteachers have doctorates in education. The building studied enrolls morestudents than the other four K-4 buildings in the district and alsoenrolls the highest percentage of African American students. All theother elementary schools have an African American student populationless than 50 percent. About one-half (approximately 150 students) of theAfrican American students in this school come from middle classfamilies. The other half (150 students) of the African American studentsare from lower middle class or economically disadvantaged This article or section may contain original research or unverified claims.Please help Wikipedia by adding references. See the for details.This article has been tagged since September 2007. families.Ninety percent of the potentially at-risk students The term at-risk students is used to describe students who are "at risk" of failing academically, for one or more of any several reasons. The term can be used to describe a wide variety of students, including, ethnic minorities academically disadvantaged are lower middleclass or economically disadvantaged African American students.In 1990, school principals (principal and assistant principal, whois now principal in another K-4 elementary school in the same schooldistrict) and some teachers led an effort to introduce socialconstructivist theory in most classrooms. State testing began in 1995.Most of the questions on the fourth grade mathematics test were multiplechoice (35 out of 40). Only five questions asked students for writtenexplanations for solving problems. Nearly all test questions askedstudents to apply skills in the context of relevant mathematicssituations.Educators could not easily obtain copies of the test booklets ortheir students' test responses. Educators were told that the statewas not equipped to release classroom sets of students' testbooklets. To obtain a specific students test booklet, educators/parentshad to write a request and pay to have the test booklet copied (bookletsaveraged $8.00-$9.00 each). Obtaining a copy of a students booklet wasnot easy. For example, in July July:see month. 1999, the principals requested 25booklets of underachieving students. They wanted to analyzestudents' problem-solving problem-solvingn → resoluci��n f de problemas;problem-solving skills → t��cnicas de resoluci��n de problemasproblem-solvingn → strategies. State officials werereluctant to provide the testing booklets when informed that informationfrom the test was being used for this paper.State Official: We can't release these booklets for research.Students booklets are confidential.Principal: But I am their principal who happens to be conducting aresearch study about my underachieving students' ability to performon the proficiency test proficiency testn → prueba de capacitaci��n. How am I going to improve instruction withoutknowing how these kids did? That is what the state wants, to improvestudent learning and get all these kids through the test, right?State Official: Well, I need a letter from your superintendent thathe is aware of this research study and will guarantee that you havetaken precautions precautionsInfectious disease The constellation of activities intended to minimize exposure to an infectious agent; precautions imply that the isolation of an infected Pt is optional, but not mandatory. about confidentiality of subjects. (Phoneconversation, mid-July n. 1. the middle part of July.Noun 1. mid-July - the middle part of Julyperiod, period of time, time period - an amount of time; "a time period of 30 years"; "hastened the period of time of his recovery"; "Picasso's blue period" , 1999)Educators at this school faced the political pressures of statetests. The publication of test results disturbed the climate of thislearning community. In 1998, only 69% of fourth grade students passedthe mathematics tests, which was clearly below the 80% average passagerate of other elementary schools in the district. Because of testresults, some parents, teachers, school board members, and centraloffice administrators questioned the credibility of the mathematicsreform. School principals and teachers searched for ways to blend socialconstructivist practices with preparation for the state mathematicstest. It's a focused curriculum. Focused on big ideas and on the skills necessary to take those big ideas and apply them to real world situations. The problems that we give the kids are highly contextual. So if you are asking me what makes this different from what you might see in other classrooms, I would think that in other classrooms, mathematics is driven by textbooks. There is no math textbook here (former assistant principal and current principal of another school in the same school district).The reform tacked the challenge of closing the achievement gapbetween white and African American students on the state test. InSeptember September:see month. 1998 the school principals extended instructional time with"at-risk at-riskadj.Being endangered, as from exposure to disease or from a lack of parental or familial guidance and proper health care: efforts to make the vaccine available to at-risk groups of children." students. About 45 third graders and about 40 fourthgraders (40% of each grade level) attended extended instructionalprograms. Instruction was designed 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. constructivist learningtheory. Main mathematical topics were emphasized as outlined in NCTMStandards (1989, 1991, 1995). Instruction was intentionally in��ten��tion��al?adj.1. Done deliberately; intended: an intentional slight.See Synonyms at voluntary.2. Having to do with intention. active andinteractive with learning experiences that stressed writing, speaking,illustrating, building, and role-playing role-play��ingn.A psychotherapeutic technique, designed to reduce the conflict inherent in various social situations, in which participants act out particular behavioral roles in order to expand their awareness of differing points of view. mathematical problems Mathematical problem may mean two slightly different things, both closely related to mathematical games:general meaninga question that can be answered with the help of mathematics ; formal meaning : any tuple (S, C( ), r . Beforestudents took the test in March, fourth grade scholars received about240 extra hours of mathematics tutoring. Parents were urged to enrolltheir children in these extended programs. As more and more parentsnoticed and heard about the popularity and effectiveness of thesetutoring programs, enrollment grew and most parents praised theschool's efforts. Potentially at-risk students were selected andinvited based on scores from teacher-made tests, classroom performance,and scores on a third grade Standford Achievement Test (SAT). Inaddition to closing the achievement gap, the reform confronted thechallenge of teaching high level mathematics to all students so thatstudents and educators could meet the goals of a postmodern post��mod��ern?adj.Of or relating to art, architecture, or literature that reacts against earlier modernist principles, as by reintroducing traditional or classical elements of style or by carrying modernist styles or practices to extremes: society. I think it's a shift in the way you think about yourself as a teacher. I think the biggest challenge is getting people to want to devote more time and more energy to improving mathematics education (principal).Parents actively supported the framework of social constructivistteaching theory for mathematics learning even when test scores were low.Parent support, confidence, and enthusiasm increased when 1999mathematics scores showed a dramatic improvement (69% passage rate to a90% passage rate). Mathematics scores continued to improve over the nextfour years, and the achievement gap continued to narrow.Theoretical and Philosophical AssumptionsThe theory and philosophy of this study were influenced by socialconstuctivist epistemology epistemology(ĭpĭs'təmŏl`əjē)[Gr.,=knowledge or science], the branch of philosophy that is directed toward theories of the sources, nature, and limits of knowledge. Since the 17th cent. . Social constructivist theory posits thatlearning and knowing are build via active and interactive activities ina classroom (Bauersfeld, 1988; Cobb & Yackel, 1996). This theoryvalues time for discourse among members of the learning community andtime for building or drawing models of mathematical situations (Fennema,Franke, Carpenter, & Carey, 1993; Simon, 1995). Furthermore, thetheory recognizes prior and present experiences, relevancy of context,and the value of multiple perspectives (Guba GUBA Gigantic Usenet Binaries Archive & Lincoln Lincoln, city and district, EnglandLincoln,city (1991 pop. 79,980) and district, Lincolnshire, E England, in the Parts of Kesteven, on the Witham River. , 1994).Social constructivist theory assumes a teaching/learningenvironment beyond the rote rote?1?n.1. A memorizing process using routine or repetition, often without full attention or comprehension: learn by rote.2. Mechanical routine. and routine learning of basic mathematicsskills, not in place of learning basic skills. Social constructivistteaching practices emerged as an important feature for students'understanding of mathematics beyond the limited memorization mem��o��rize?tr.v. mem��o��rized, mem��o��riz��ing, mem��o��riz��es1. To commit to memory; learn by heart.2. Computer Science To store in memory: of basicfacts and mechanical procedures. In this sense, classroom practicesfocus on: dialogue, prior knowledge, mathematical modeling Note: The term model has a different meaning in model theory, a branch of mathematical logic. An artifact which is used to illustrate a mathematical idea is also called a mathematical model and this usage is the reverse of the sense explained below. , multiplesolutions, students' preconceptions, problem solving problem solvingProcess involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error. , problemposing, and the importance of context for building understanding(Romberg, Shafer, & Webb, 2000).Methods and Study DesignThe study was grounded in the constructivist inquiry of Guba &Lincoln (1989, 1994) and Lincoln & Guba (1985). It was anobservational study In statistics, the goal of an observational study is to draw inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator. . Participant-observation, interviews, and review ofpublic documents were used to collect information and to interpret thedata. Interview data included the transcription transcription/trans��crip��tion/ (-krip��shun) the synthesis of RNA using a DNA template catalyzed by RNA polymerase; the base sequences of the RNA and DNA are complementary. tran��scrip��tionn. of audio tapes frominterviews with teachers, principals, assistant superintendents Assistant Superintendent, or Assistant Superintendent of Police (ASP), was a rank used by police forces in the British Empire. It was usually the lowest rank that could be held by a European officer, most of whom joined the police at this rank. ,superintendents, community officials, parents, and students. Field noteswere used when the primary researcher observed and participated in theteaching/learning process. Data were collected over a five-year period(1999-2003). Additional data sources were triangulated and negotiatedamong the researchers (one university teacher and two school principals)for trustworthiness trustworthinessEthics A principle in which a person both deserves the trust of others and does not violate that trust of data analysis. Based on multiple data sources,several themes emerged, and categories were developed. In what follows,we discuss each of these themes.Lessons LearnedMajor mathematics reform at the elementary level probably requiresstructural and cultural changes. Implementing social constructivistpractices in elementary mathematics Elementary mathematics consists of mathematics topics frequently taught at the primary and secondary school levels. The most basic are arithmetic and geometry. The next level is probability and statistics, then algebra, then (usually) trigonometry and pre-calculus. classrooms may call for changingconventional teaching, learning, and assessment practices. I taught, years ago, in a very traditional classroom. And, I taught exactly how my advising teaching taught. She introduced me to a college professor who was doing math differently. I had been a very apprehensive math student, but she was teaching the [math] and showing it through games, manipulative, and stuff. It was fascinating to me. I lapped up everything I could. I'd tell the kids that I didn't understand this [mathematics] when I was their age. I didn't have a clue what mathematics was. It's kind of neat! (fourth grade teacher).Most parents and teachers did not learn mathematics in a settingwhere instruction focuses on the development of mathematical concepts.Many teachers and principals may need to relearn Verb 1. relearn - learn something again, as after having forgotten or neglected it; "After the accident, he could not walk for months and had to relearn how to walk down stairs" mathematics. If you are going to apply it [social constructivism], if you're going to build and draw models, role play it, and have time to correct and discuss misunderstandings, change your mind and rebuild things, and come back and revisit things, and try to put it all together, that takes time (principal).Some parents and concerned citizens may also need to viewmathematics education differently. Most school schedules (day and year)do not provide enough time for the effective implementation of socialconstructivist practices. Classrooms are often not equipped withresource materials that support active, interactive instruction. Someschools which serve lower middle class and economically disadvantagedstudents may need more technology to support changes in mathematicseducation. In what follows, we discuss structural changes, culturalchanges, politics of reforming mathematics education, and AfricanAmerican students' engagement in a social constructivist setting.Structural ChangesTeaching all students to understand and apply key mathematicalideas within a social constructivist framework necessitates extendinginstructional time. Some students need extended time to learn andunderstand mathematics. Mathematics classroom instruction at this K-4elementary school was extended from 45 minutes to 90 minutes for allstudents each day.Students who were still underachieving at the 3rd and 4th gradelevel were invited to participate in an extended school day, week, andyear. Morning and afternoon tutoring was provided (75 minutes beforeschool for four days each week, 45 minutes after school for three dayseach week). Teachers and principals tutored the students. Tuition-freesummer school was offered for six weeks, and three and a half hours ofinstruction was provided on each Saturday Saturday:see week; Sabbath. from September to middle ofMarch. Students who participated in extended time initiatives had about240 more hours of instruction prior to taking the state mathematicstest. About 90% of participating students in extended time programs wereAfrican Americans whose parents drove them to school early each Saturdaymorning and early to school for four days a week. Some parents did nothave access to transportation on Saturday mornings, so the programcoordinator drove a car to pick up about six students every Saturdaymorning.Most instruction during these extended sessions was conductedwithin a social constructivist framework. Computers and mathematicssoftware programs were used extensively to support classroominstruction. Social constructivism constructivism,Russian art movement founded c.1913 by Vladimir Tatlin, related to the movement known as suprematism. After 1916 the brothers Naum Gabo and Antoine Pevsner gave new impetus to Tatlin's art of purely abstract (although politically intended) in practice demanded moreinstructional time because it went beyond basic calculation skillstowards an emphasis on understanding how calculation skills are used inreal-life real-life?adj.Actually happening or having happened; not fictional: a documentary with footage of real-life police chases.contexts. Seeing how you ask probing questions and how you get students to talk about what they are seeing is my favorite part. It's really not only teaching me about math, but also teaching me how I can be a better teacher (first year teacher's reflection on teacher- leader's interaction with students involved with extended time programs).The school did not adopt a constructive pedagogy completely. TheKumon Mathematics Program was in use as a tutoring tool and wasavailable to all students who were at-risk based on low scores fromteacher-made basic computation Computation is a general term for any type of information processing that can be represented mathematically. This includes phenomena ranging from simple calculations to human thinking. tests. The Kumon Mathematics Program usesa behaviorist Behaviorist1. One who accepts or assumes the theory of behaviorism (behavioral finance in investing.) 2. A psychologist who subscribes to behaviorism.Notes:When it comes to investing, people may not be as rational as they think. approach for learning basic arithmetic skills. The programcosts about $30.00 per student. About eight of the schools'teachers used Kumon materials for 45 minutes three afternoons each weekfrom January January:see month. of the students' third year until April of thestudents' fourth year. Overall, the program costs the districtabout $17,000 per year. Kumon was used in after school tutoring sessionsand offered free of charge to students. The researchers were uncertainthat all students could effectively learn basic facts within a socialconstructivist framework. Also, the researchers were aware that therewas little time to teach both basic skills and conceptual understandingin order to improve students' achievement on the state tests. TheKumon Program was done after school and away from regular classroom timeso that teachers could focus on learning mathematics within aproblem-solving and problem-finding environment.Cultural ChangesUsing a learning theory that moved away and beyond the behaviorism behaviorism,school of psychology which seeks to explain animal and human behavior entirely in terms of observable and measurable responses to environmental stimuli. Behaviorism was introduced (1913) by the American psychologist John B. created disequilibria for some teachers, some district administrators,some parents, and some members of the school district's boards ofeducation. Mathematical processes Noun 1. mathematical process - (mathematics) calculation by mathematical methods; "the problems at the end of the chapter demonstrated the mathematical processes involved in the derivation"; "they were learning the basic operations of arithmetic" and mathematical ideas started to beviewed as less definite and more probabilistic (probability) probabilistic - Relating to, or governed by, probability. The behaviour of a probabilistic system cannot be predicted exactly but the probability of certain behaviours is known. Such systems may be simulated using pseudorandom numbers. than most parents andteachers had learned when they were young. Most teachers learned moreabout mathematics and the application of mathematical skills andconcepts. Some teachers began to view mathematics as a creative process.They taught students to create different patterns, to understandprobability, and to design different strategies for solving mathematicalproblems. It's the leadership of the building, number one. There is a real commitment to the fact that all students can learn. There is not a student at this building that can't learn. I see an emphasis on the extra time that the students need to learn as a big priority at this school. They are doing whatever it takes here to get it done. A lot of parents almost come to expect it. I also see a focus on curriculum a lot more intense here than at some of the other buildings [district schools] in terms of what the most up-to-date research says about how we're teaching. Some of those practices are put into play here at this building (new assistant principal of the school).Many parents learned to appreciate, enjoy, and view mathematics asa creative art. Many parents also respected their child's use ofdiscourse, illustration, and concrete models to build understandingabout important mathematical concepts.Perhaps the most interesting lesson learned from this reform effortwas the importance of "student voice" in mathematicaldiscourse. Discourse and "student voice" providedopportunities for students' self-reflection self-re��flec��tionn.Self-examination; introspection.self-re��flec and opportunities forteachers to understand what students did not know and what to do next.This recursive See recursion. recursive - recursion relationship between "student voice" andclassroom discourse was pivotal for reforming instruction andcurriculum. Mathematics is hard and sometimes it can be fun but it is hard. I want to learn mathematics but also I want to have fun while I do mathematics. I have to think a lot while I am problem solving at [name of school]. We have to explain our reasoning. We have to draw pictures of money when we are solving problems. We do a lot of discussions while we are solving fraction problems. I like fractions and I like graphing. It is fun to put stuff on the chart and color it in to measure ... There is not one way to answer a question, there is always another way you can do it. In problem solving you have to use your mind, you have to go way back when you explain, you have to use all your knowledge (fourth grade student).Guided by "student voice" and classroom dialogue,teachers realized that textbook textbookInformatics A treatise on a particular subject. See Bible. instruction and standardized standardizedpertaining to data that have been submitted to standardization procedures.standardized morbidity ratesee morbidity rate.standardized mortality ratesee mortality rate. courses ofstudy might be contextually limited. It seemed that value placed on"student voice" and student discourse provided a more open,respectful re��spect��ful?adj.Showing or marked by proper respect.re��spectful��ly adv. , and democratic environment for the mathematics classroom."You 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. what they're they're?Contraction of they are.they'rebe thinking unless you askthem. I only get it [instructional decisions] from the voice of thestudents. They greatly impact the instructional path that I take"(former assistant principal of the school).Social Constuctivism and African American StudentsIn 2001, 100% of the elementary school's white students and92% of the school's African American students passed thestate's fourth grade mathematics test. The difference betweenAfrican American scores and white scores was much smaller than thedifferences between these two racial groups' scores at the locallevel which was 98% passage rate for white and 70% passage rate forAfrican American. One might conjecture CONJECTURE. Conjectures are ideas or notions founded on probabilities without any demonstration of their truth. Mascardus has defined conjecture: "rationable vestigium latentis veritatis, unde nascitur opinio sapientis;" or a slight degree of credence arising from evidence too weak or too that social constructivistteaching practices within an extended instructional program caneffectively raise African American achievement in mathematics. Theresearchers suggest that extended time is required because socialconstructivist practices such as dialoguing, role-playing, building,illustrating, and writing do not fit into the framework of conventionalschool schedules. The principals also conjecture that the Kumon programwhich is a behaviorist approach to learning mathematics was a necessarycomponent for building a basic foundation for such students.In what follows, we demonstrate a small group problem solvingsituation during a Saturday morning class in spring 2003. Forty-fivestudents participated in the Saturday program. The students were dividedinto small groups of six, seven, eight, nine, and ten students. Someteacher-leaders and principals agreed to work with these students intheir small groups. The following episode is about a fourth gradeteacher's interaction with six African American students. Theteacher was sitting in a low chair and the six students were sitting onthe carpet floor in a half circle. The teacher posed a problem aboutfractions.Teacher: Here is the problem. If Olivia Olivia“abjured the company and sight of men.” [Br. Lit.: Twelfth Night]See : Isolation ate one and six eighths ofsome candies [she wrote on a small, white erasable e��ras��a��ble?adj.1. Capable of being erased: erasable ink.2. Capable of producing something that can be erased: an erasable pen. board O = 1 6/8].Brianna “Briana” redirects here. For the fish species, see Bri��na.Brianna (also commonly spelled Briana) is an English name. It is the modern English feminine form of Brian. "Briana" originated from Ireland and means "Strong will of God". ate three eighths [she wrote B = 3/8]. Latasha ate one and onefourth [she wrote L = 1 1/4]. Cierra ate half [she wrote C = 1/2].Marcus Marcus,in the Bible: see Mark, Saint. ate four eighths [she wrote M = 4/8] and Wesley ate three fourths[she wrote W = 3/4]. I want you to figure out how much candy candy:see confectionery. candySweet sugar- or chocolate-based confection. The Egyptians made candy from honey (combined with figs, dates, nuts, and spices), sugar being unknown. they allate [students were listening very attentively]. We want to combine allthe candies they ate together. Now, your job is to draw nice and neatpictures and explain to us how you figured out the solution. Go for it!Students started tackling the problem individually and quietly bydrawing pictures of candies and cutting them into halves, fourth, andeighths. The teacher carefully observed students' illustrations.After about 15 minutes, she asked each student to demonstrate his/hersolution. Students and teachers established norms that required studentsto listen to each other's explanations, to agree or disagree with Verb 1. disagree with - not be very easily digestible; "Spicy food disagrees with some people"hurt - give trouble or pain to; "This exercise will hurt your back" each other, and to ask questions if they did not understand theirpeer's solutions. The teacher would facilitate the activity byobserving, listening, and asking guiding questions.Teacher: Okay, who wants to draw a picture on the board and showhow many candies Olivia ate? Latasha!Latasha: [She went closer to the white board and drew one wholecandy and divided another candy first in halves See In half , then in fourths andeighths.] Olivia ate one whole candy and six eights. [She colored onewhole candy with blue marker marker/mark��er/ (mahrk��er) something that identifies or that is used to identify.tumor marker and six eighths of another whole candy withred marker.]Teacher: Do you all agree with Latasha's solution? [Studentsnodded their heads in agreement.] Now, I want a volunteer to show howmany candies Brianna ate.Marcus: [He went down to the board and picked up a blue marker.]Brianna ate three eights. I divided a whole candy into eight equalpieces and colored three pieces. [He drew a picture for his solution.Students agreed with his solution.]Teacher: Brianna, you are very quiet. Are you sleepy sleepycharacterized by sleep.sleepy foal diseasesee shigellosis.sleepy staggerssee hepatic encephalopathy. ?Brianna: No!Teacher: Can you show me how many candies Latasha ate?Brianna: [With a strong voice] yeah. [She picked up a brown markerand drew one whole candy and one fourth of another candy. Then shecolored 1 1/4]. My answer is one and one fourth.Wesley: I can show that another way. [He went close to the whiteboard and drew one and two eighths and colored them.]Teacher: Do you all agree with Wesley's solution?Students: Yeah [almost unanimously].Teacher: Okay, who wants to go next? Cierra?Cierra: I ate half of a candy and that is the same as Marcus'four eighths because one half equals four eighths.Teacher: Wow! I am impressed im��press?1?tr.v. im��pressed, im��press��ing, im��press��es1. To affect strongly, often favorably: . Can someone show me three fourths inanother way?Brianna: Three fourths is the same as six eighths because I startedwith a whole candy and I divided it in half then I divided it in fourthsbecause the whole piece is four, and I colored three pieces of fourpieces. Then I divided the four pieces into eight pieces and I foundthey are equal. [Students agreed with Brianna's solution.]Teacher: Now, who can figure out the most and least candy bareaten?Olivia: I ate the most and Cierra ate the least.Latasha: I disagree, because Cierra's half candy is foureighths and Brianna ate three eighths.Olivia: Oh, that's right, Brianna ate the least.Teacher: Olivia, you ate a lot. Brianna was not very hungry. Now,who can show how many candies were eaten all together?Latasha: I think all of the candies eaten were five.Teacher: Do you all agree with Latasha's solution?Students: Yeah, no, yeah.Teacher: Okay, we have differences of opinion. Now how many of youagree with Latasha's solution, raise your hands. [Two studentsraised their hands.] Okay, how many of you do not agree withLatasha's solutions, raise your hands. [One student raised herhand.] How many of you aren't aren't?Contraction of are not. See Usage Note at ain't.aren'tare notaren'tbe sure? [Two students raised theirhands]. Okay! Cierra, you don't don't?1. Contraction of do not.2. Nonstandard Contraction of does not.n.A statement of what should not be done: a list of the dos and don'ts. agree with Latasha's solution.Would you come up here and show us how you figured out your solution?Cierra: [Quietly went closer to the white board and explained hersolutions.] Well, I know that I ate half and Marcus ate half, sothat's one. Then I know that Olivia ate one whole and six eights,and Latasha ate one whole and two eighths, that is three whole. [Shedrew her half and Marcus's half as one whole with two differentcolors, and shaded Olivia and Latasha's as three whole with adifferent color.] I know Brianna ate three eights and Wesley ate sixeighths, that is one more whole candy and one eighth left over.[Students were carefully listening.]Teacher: Do you all agree with Cierra's solution?Marcus: I don't get it. Where did you get one eighth?Latasha: I know where I went wrong. It is five whole and oneeighth. I didn't did��n't?Contraction of did not.didn'tdid notdidn'tdo count the fraction correctly. Six eighths andthree eighths would be one and one eighth.Marcus: But three eighths and six eighths are nine sixteenths! [Hemade sixteen pieces and colored nine of them.]Brianna: But you don't have to make sixteen pieces. You needto count three pieces of a whole and sixth pieces of another whole. Thatmakes it one and one eighth left over.Marcus was puzzled puz��zle?v. puz��zled, puz��zling, puz��zlesv.tr.1. To baffle or confuse mentally by presenting or being a difficult problem or matter.2. and needed more time and experiences withfractions. The teacher recognized that she needed more one-on-one one-on-oneadj.1. Consisting of or being direct communication or exchange between two people: one-on-one instruction.2. Sports Playing directly or exclusively against a single opponent. interaction with Marcus regarding the relationship between parts andwholes.The above episode demonstrates the social norms established andaccepted by teachers and students. The students were expected to listenand respectfully re��spect��ful?adj.Showing or marked by proper respect.re��spectful��ly adv. challenge each other's solutions. They wereobligated ob��li��gate?tr.v. ob��li��gat��ed, ob��li��gat��ing, ob��li��gates1. To bind, compel, or constrain by a social, legal, or moral tie. See Synonyms at force.2. To cause to be grateful or indebted; oblige. to provide support for each other and communicate their sharedmeaning with one another. The teacher's role was to posechallenging problems and help students reach their zone of potentialdevelopment. The climate of the classroom was conducive con��du��cive?adj.Tending to cause or bring about; contributive: working conditions not conducive to productivity.See Synonyms at favorable. to learning. Theteacher encouraged students' risk-taking in a problem solvingsituation. She valued and reflected on Marcus's situation of"not having figured it out yet." Due to time constraints In law, time constraints are placed on certain actions and filings in the interest of speedy justice, and additionally to prevent the evasion of the ends of justice by waiting until a matter is moot. , shecontinued her dialogue with Marcus after the class period.The next episode illustrates student-principal interactions in theSaturday morning program where the focus was on key ideas inmeasurements and fractions. In this episode, one of the principals foundthat the students in her Saturday class had limited understanding of thepassage of time. Ten African American students and the principal sattogether in a circle on the floor. A large pad of newsprint newsprintlow grade paper used for newspapers. Old newspapers are fed to cattle as an alternative roughage and may occasionally be ingested by dogs. Significant amounts of lead are accumulated in tissues; no cases of poisoning have been recorded in cattle, though it has been , markers, adry erase floor easel, a box of play money, and a large "Judy"clock were available materials. In this setting, she designed thefollowing problem to address students' preconception pre��con��cep��tion?n.An opinion or conception formed in advance of adequate knowledge or experience, especially a prejudice or bias.Noun 1. of elapsedtime e��lapsed timen.The measured duration of an event.Noun 1. elapsed time - the time that elapses while some event is occurring .Principal: OK, I have a tricky Adrian Thaws (born January 27, 1968), better known as Tricky, is an English rapper and musician important in the trip hop and British music scene (despite loathing the "trip hop" tag). He is noted for a whispering lyrical style that is half-rapped, half-sung. problem for you to solve today.Student A: You can't trick us. We know everything!Student B: Yeah, you haven't been able to trick us yet.Principal: But, remember, I am not paid to teach you what youalready know. I get paid to 'mess with your minds,' confuse con��fuse?v. con��fused, con��fus��ing, con��fus��esv.tr.1. a. To cause to be unable to think with clarity or act with intelligence or understanding; throw off.b. you. So, you better watch out because today I am going for 'bigbucks!' Now, here is the problem. Watch me as I illustrate it onthe white board. [Principal draws a happy face on the board. Childrengiggle at the hair.] Ok, the hair looks a little strange but what shouldyou be doing?Student C: Listening to the problem and seeing what importantinformation is in it.Principal: [Continuing to draw and dictate TO DICTATE. To pronounce word for word what is destined to be at the same time written by another. Merlin Rep. mot Suggestion, p. 5 00; Toull. Dr. Civ. Fr. liv. 3, t. 2, c. 5, n. 410. problem] Jason wants toearn money to buy a bike. To earn money for the bike, Jason takes careof pets. He charges $4.00 an hour. [She draws four-one dollar billsunder a picture of dogs and cats] Mr. Johns wants Jason to take care ofhis puppy puppythe young of the canine species; usually used up to the age of 12 months.fading puppy syndromesee fading kitten/puppy syndrome.puppy pyodermasee impetigo. from 1:30 p.m. until 4:00 p.m. How much money will Jason earnto take care of Mr. Johns' little puppy?Student C: How much is the bike?Principal: I don't know. What do you think? Is it a part ofthe information needed to solve the problem?Student D: No, not really. The question did not ask about the bike,only about how much money would Jason earn from Mr. Johns.Principal: Does everyone agree with [name of student D]'sanswer?Student E: Yeah, I guess you don't need the cost of the bike.But the price is important because Jason doesn't know how long hewill have to watch dogs in order to have enough money.Principal: That's a good thought. I understand that you wantto know the price and leaving it out bothers you. But, do we need thecost of the bike?Student E: No.Principal: Can we go on to solve this time and money problem?[Students nod their heads, answer, 'yes'].Principal then asks students to retell re��tell?tr.v. re��told , re��tell��ing, re��tells1. To relate or tell again or in a different form.2. To count again.Verb 1. the problem and identify theimportant information needed to solve the problem. Students are givenabout ten minutes to solve the problem. Students are encouraged to worktogether, compare their solutions. After the ten minutes, the principalreconvenes the discussion:Principal: OK, who wants to tell us how much money Jason earned?Who wants to share their strategy and thinking?Student F: I will.Principal: Ok, [student's name] move up here, next to theboard and show us how you solved the problem. [Student moves next to thewhite board easel.]Student F: First, the answer is $14.00.Principal: Does everyone agree with this answer. [Most studentsagree, Student G does not agree.]Student G: I got $10.00.Principal: Any other solutions?Student H: I got $11.00.Principal: Other answers? [There are no other answers.]Principal: Ok, so what do we know? Can all the answers be correct?Student A: No, because this is math and there is usually only oneright answer.Principal: [laughing], Not always but maybe in this case. We willhave to find out if we have more than one 'right' answer. Youguys know how I feel about 'one right answer' in math. Itisn't about the answer but the way you think about the answer.Students: Yeah, we remember.Principal: Ok, [calls student F's name], share your thinkingwith us. How did you arrive at your solution?Student F: Well, Jason makes $4.00 an hour--just like your pictureon the board. He works for 3 and 1/2 hours. Three hours times 4 is 12.Principal: Why did you multiply mul��ti��plyv.1. To increase the amount, number, or degree of.2. To breed or propagate. by four?Student F: [Drawing three circles on the board] These are hours.For every hour, Jason earns four dollars [student puts $4.00 inside eachcircle.] You can add 4 + 4 + 4, or you could just multiply 3 times 4 andget the answer which is 12.Student B: But you said the answer was 14.Student F: I am not finished. Then, Jason works more than 3 hours.He works 30 minutes more-until 4:00. So, then thirty minutes is 1/2 ofan hour, so I halved halve?tr.v. halved, halv��ing, halves1. To divide (something) into two equal portions or parts.2. To lessen or reduce by half: halved the recipe to serve two.3. the four dollars. That is two dollars. I added thetwo dollars to 12 and got 14. So that's that. [Student F returns toplace in the circle. There is a silence as students mull over mull overVerbto study or ponder: he mulled over the arrangements[probably from muddle]Verb 1. theinformation.]Principal: Well, what are you thinking? I smell smoke, so some ofyou are really thinking about [Student F's name] solution.Student G: Well, I am changing my answer. I agree with [studentF's name]. I forgot to add the money for the half hour.Principal: OK, you changed your answer. You found a mistake, good.What about you, [calls student H's name]? What do you think?Student H: Well, now I am not sure. When I did it, I thought that Ihad the answer, now, I don't know.Principal: Why don't you show us your solution? [Student Hmoves to the easel.]Student H: I think Jason earned $10.00 because he worked 2 1/2hours, not 3 1/2.Principal: How did you figure it was 2 1/2 hours? [Student Hreaches for the clock and sets the time for 1:30.]Student H: [Moving hands of the clock] 1:30 to 2:30 is one hour.2:30 to 3:30 is another hour. 3:30 to 4:00 is thirty minutes or 1/2hour. So Jason earned $10.00. I figured the money just like [studentF's name] except he had Jason working one more hour. I am not sureif I counted the hours right.Principal: What do all of you think? We have a difference of onehour. How should we solve this difference of hours Jason worked?Student I: [Calls student F], show me how you counted the hours.Student C: Yeah, use the clock, like [student H]Student F: Ok, watch. [Student F takes the clock and moves thehands.] Begin at 1:30-that's one hour, 2:30, that's 2 hours,3:30-3 hours, 4:00-thirty minutes more. 3 1/2 hours. [Again, studentsare silent.]Student H: But you counted 1:30 as one hour. I don't think youcan do that.Student F: What do you mean?Student H: Well, when we come to school at 9:00. We haven'tbeen there even one hour. Jason shows up at 1:30 but he has not workedyet. He begins to work. When he has worked until 2:30, then he hasworked for one hour. Just like school. If your teacher says that in onehour, we will go to art, you know that at 10:00 you have art. Youdon't go to art at 9:00. [Now other students are listening tostudent H's argumentation.]Student D: Maybe [student H] is right. One hour has to go by beforeyou can say that you worked one hour. [Student H is getting more supportfrom classroom community.]Student E: Yeah, I know that my baby-sitter doesn't get paidwhen she comes to the house. She has to work first.Principal: Your thinking is interesting. Let's see Let's See was a Canadian television series broadcast on CBC Television between September 6, 1952 to July 4, 1953. The segment, which had a running time of 15 minutes, was a puppet show with a character named Uncle Chichimus (voice of John Conway), which presented each if we canact this problem out.Students: Yeah, this will be fun!Principal gets several sheets of paper. She places the papers in aline. There is a 20 inch distance between each paper. She instructs thestudents to stand and gather around papers.Principal: Ok, let's let each paper stand for the time thatJason worked. What time did Jason begin the job?Students: 1:30. [Principal writes 1:30 on one paper.]Principal: And one hour later?Students: 2:30. [Principal writes 2:30 on the next paper in line.]Principal: And one hour later?Students: 3:30. [3:30 is written on the next paper.]Principal: And one hour later?Students: 4:30. [4:30 is written down.]Student E: But Jason only worked until 4:00. You need to change it.Principal: Thank you, I made a mistake. See, I was trying to trickyou. Ok, who wants to be Jason? [Several students volunteer.] Ok, [callsstudent C's name], you be Jason and let's act out [student F]solution. I will be Mr. Johns. Let me get some play money for this mathdrama. [Principal takes a handful of one dollar bills from the box onthe floor.] OK. Jason, thank you for coming over to watch Little Rover.Here's $4.00. [Principal hands Student F four dollars.]Students: No, that's not right.Principal: What do you mean?Student A: Don't give him money, yet. He hasn't watchedRover. He just got there.Principal: Let's just continue to use [student F's]strategy. Let me continue to pay Jason, then we will look at [studentH's] strategy and then we will discuss them. Ok, Jason, move to2:30. [Student steps on next paper.] Good, here's your next fourdollars. Now move to 3:30. Ok, here's another four dollars. Now,what do we do? Jason did not work until 4:30?Student I: Have [student C] stop in the middle, the space between3:30 and 4:30. That will show 1/2 hour.Principal: Does that make sense to everybody?Students: Yes.Principal: So how much money do I give Jason?Student A: $2.00.Principal: Why?Student D: Because 2 dollars is half of four dollars. [Principalhands 'Jason' 2 more dollars.]Principal: OK, 'Jason,' count your money.Student F: I have $14.00.Principal: $14.00. You all saw it. It must be the right answer,right?Student H: We didn't do my answer.Principal: That's right. Why don't you be Jason? Begin at1:30. [Student stands next to the paper labeled 1:30.]Principal: Hi, 'Jason' Welcome. Little Rover is happythat you've come to watch him. Here's your $4.00.Student H: No, I haven't worked an hour yet. I don't getpaid right now. See, you have to work to get paid. Watch, I show up at1:30. Mr. Johns leaves. I play with Rover for 60 minutes. [Student movesto the paper marked 2:30.] 2:30 is one hour past 1:30. 60 minutes hasgone by. Now I can be paid 4 dollars. [Principal gives four dollars tothe student.] Then, I play another sixty minutes. 2:30 and 60 minutesgoes by. That makes 3:30, another hour. Now, pay me again because Iworked another hour. [Principal gives student another four dollars.]Principal: How many hours has this 'Jason' worked so far?Students: Two. You only paid him for two hours.Student H: OK. Now watch. I don't play with Rover until 4:30because Mr. Johns comes back at 4:00. From 3:30 to 4:00 is only 30minutes. One half hour. I don't work a whole hour. I don't getanother four dollars. I get half of four dollars. Half of four dollarsis two. So now, you can give me two more dollars. [Principal givesstudents two more dollars.]Student H: Now, I am going to count my money. 2, 4, 6, 8, 10dollars. Yup, Jason made ten dollars.Principal: OK. We have two different solutions for this problem.What do you think?Student F: I think I want to change my answer. [Student H] isright. You have to work for an hour to get paid for an hour. I think Iknow how I made the mistake about hours. I went like this. [Studentholds up his hand.] 1:30 [holds up one finger.] That's one, 2:30[holds up another finger.] That two, 3:30 [holds up a third finger.]That's 3 and 3:30 to 4:00 is half.Principal: That's a good demonstration. How would you changeyour counting of the hours?Student F: Maybe what I could do would be to say quietly, 1:30 butdon't count it. I would begin counting with 2:30. Time has to go bybefore you count it.In the above episode, we observed three components ofproblem-centered inquiry: communication, questioning strategy, and useof manipulative ma��nip��u��la��tive?adj.Serving, tending, or having the power to manipulate.n.Any of various objects designed to be moved or arranged by hand as a means of developing motor skills or understanding abstractions, especially in (I.e., a large pad of newsprint, markers, a dry erasefloor easel, a box of play money, and a large "Judy" clock).The problem presented by principal was interesting. It was within thestudents' zone of potential construction. Students wanted to findout a viable solution. Communication between the principal and studentswas focused on mathematical argumentations and justifications. Theprincipal did not dominate the conversation. Instead, she providedstudents with opportunities to explain their thinking and reasoning.Students were expected to listen carefully to one another and askchallenging questions. When the principal asked questions, she intendedto understand students' thinking and guided them toward makingsense of mathematics. She provided students enough time for reflectionsand modifications of their solutions. The learning climate supportedstudents' inquiry.Another important characteristic of the problem-centered inquiryapproach was the principal's and students' role playing role playing,n in behavioral medicine, learning exercise in which individuals assume characters different from their own. The individual may also be asked to simulate a particularly difficult situation and apply the characteristics that are common to his . Roleplaying and use of manipulative provided the classroom community acontext that was relevant to them. Within the context, they were able tomodel and interpret the mathematical situation. Students were encouragedto restate re��state?tr.v. re��stat��ed, re��stat��ing, re��statesTo state again or in a new form. See Synonyms at repeat.re��state the problems and communicate their solutions verbally andpictorially pic��to��ri��al?adj.1. Relating to, characterized by, or composed of pictures.2. Represented as if in a picture: pictorial prose.3. .African American Students' Attitudes towards Extended TimeThe principal, the assistant principal, a few teachers, and theprogram coordinator, who is also coordinator of the Kumon program,invite parents of potentially at-risk students to a meeting every Augustbefore school begins and two other meetings during the school yearbefore the state mandated mathematics test is administered in March. Thedetails of extended instructional time are explained to all parents.These details include explanations of social constructivist teachingpractices and ideas for how parents could help their sons and daughtersat home.Parents were skeptical and uncertain during the first one or twoyears of the program starting in 1996. The challenge is huge, especially for African American parents. Not all, but some, because when you're talking about having a kid come in early morning, extended during the day, and on Saturdays, of course I get a lot of complaints-saying, 'I think this is overwhelming for my child.' 'I don't think Johnny needs to come in as much.' I have to sit and sort of put the confidence in the parents and maybe sometimes I say something that, being African American, knowing that mom has to work and mom gets off work, a lot of times mom doesn't have time to sit and help the child. It's a big cultural difference. We built trust among ourselves. The parents have trust with me. They understand I have five kids who went through the district. They all graduated. They all went to college. Now, I have three grandkids here at this school. One just finished his fourth grade. He was the top of his class, going through the same program. I have to explain these to the parents that if the opportunity is there and the school is offering this help, we, as African Americans, have to take advantage of this. We can't just sit back and say, 'Hey, let's let the school do it. We'll just send our kids there and we won't do anything else.' No! It doesn't work that way. You also, as a parent, have to help in order for us to get this done! I go through this and sometimes it's frustrating to me. And, I know it's frustrating to parents, but I have to get their confidence (program coordinator & Kumon Coordinator).The first two years were frustrating frus��trate?tr.v. frus��trat��ed, frus��trat��ing, frus��trates1. a. To prevent from accomplishing a purpose or fulfilling a desire; thwart: and required formal andinformal communication between the school and home. However, because thescores rose so high (90% passage rate) in two consecutive years (1999,2000), parents' pride and confidence in the school increaseddramatically. Parents had frequent and easy access to principals and theprogram coordinator whenever they had questions about teachingschedules, schools, concepts being taught, and a student'sprogress. Parents were encouraged to call at any time if they hadquestions or concerns and all calls were returned within the same schoolday. The program coordinator made a "reminder call" to allhomes on Thursday and Friday before each Saturday morning session.Ninety percent of the African American students eagerlyparticipated in extended time programs where they learned much in aconstructivist teaching/learning environment. They also learned muchthrough Kumon mathematics program. The study evinced that studentsenjoyed learning and responded well to social constructivist teachingpractices which encouraged dialogue, risk-taking, problem-solving,writing, illustrating, concrete modeling and the use of technology.Students also responded eagerly to the Kumon program. Parents weresurprised by student achievement. In 2001, 40% of the 43 AfricanAmerican students who were considered "at-risk" in thirdgrade, scored "advanced proficient pro��fi��cient?adj.Having or marked by an advanced degree of competence, as in an art, vocation, profession, or branch of learning.n.An expert; an adept. " on the state mathematicstest. The study suggests social constructivist practices combined withKumon program were effective in raising mathematics achievement ofAfrican American students. I'll tell you what keeps me going, is I can't stand the statistics [the achievement gap between white and African American students]. I looked at my classroom list and I see more and more problems and quite frankly, I think, jeez, these are the same children who are going to be taking care of me when I'm older. And if I can't change this trend what is going to happen to these kids? (fourth grade teacher)This fourth grade teacher's statement reminds us ofCampbell's (2001) observation of a classroom interaction. Sheshared her experience working with teachers and principals in urbansettings for instructional changes and school reform: There are many times when this effort is frustrating and the obstacles seem daunting. That is why it is important to have a shared sense of purpose with conviction. For me the source of that conviction is simple. It is the eyes of the children, children who have and more important who know they have mathematical power. (p.49)Campbell's statement echoed throughout this study from thevoices of competent and caring teachers, administrators, parents, andmost importantly Adv. 1. most importantly - above and beyond all other consideration; "above all, you must be independent"above all, most especially , young children.The problem of African American underachievement in mathematics isfilled with complexity, frustration and anger. American public educationhas been unable to effectively raise African American achievement inmathematics (Bigelow, Harvey Harvey,city (1990 pop. 29,771), Cook co., NE Ill., a suburb S of Chicago; inc. 1895. Its manufactures include steel castings, metal products, chemicals, machinery, and electronic equipment. Harvey has an oil research center. The city was founded by Turlington W. , Karp & Miller, 2001; Frankenstein,1995; Ogbu, 1987; Secada, 1992). Failure to educate African Americanstudents in mathematics limits their access to jobs in the technologicalsociety. This type of failure has economical and social justiceimplications (Lappan, 2001; Moses & Cobb, 2001; NCTM, 2000a, 2000b,2001a).Politics and ReformThe study may offer some ideas about how to raise mathematicsachievement among African American students. "The agenda isprimarily social and political ... I think that the Emmy, the Oscar, theGrammy will go to the person who can combine the social agenda and thepolitical agenda with the academic agenda" (principal).The political environment that surrounded sur��round?tr.v. sur��round��ed, sur��round��ing, sur��rounds1. To extend on all sides of simultaneously; encircle.2. To enclose or confine on all sides so as to bar escape or outside communication.n. mathematics reform atthis elementary school was challenging and somewhat explosive. Parentswere informed regularly about goals and processes for changingmathematics education. Most parents were supportive and encouraging.District administrators were somewhat skeptical and cautious. Politicscame center stage when the state began "proficiency pro��fi��cien��cy?n. pl. pro��fi��cien��ciesThe state or quality of being proficient; competence.Noun 1. proficiency - the quality of having great facility and competence " tests inmathematics for fourth grade students in 1996. The scores were used tocompare and rank elementary schools within the district and throughoutthe state. The predominately multiple choice and short answer tests werenot compatible with this school's constructivist approach toteaching and assessment.Early in the reform the fourth graders were not scoring wellcompared to fourth graders in the other elementary schools in thedistrict. However, when the principal and assistant principal startedearly morning, Saturday, after school, and summer school classes forabout one-fourth of the fourth grade students, the state mathematicaltest scores soared to the highest in the district and the state.Educators learned to maintain constructivist pedagogy within extrateaching and tutoring sessions, and students learned deeper mathematicalconcepts and scored very high.Fourth grade students achieved the following mathematics scoresover the last five years (1999-2003). In 1999, mathematics proficiencyscores where 86.1% passage rate. In 2000, the result was even better ata 90.0% passage rate. In 2001 the student's passage rate was 95.5%and 58% of the class (90 students) scored "advanced." In years2002 and 2003, the student's passage rate was 93% and 89%respectively. The five-year average was 90.7%. Because the passage ratefor African American students averaged about 85% for these five years,many parents and community leaders opined that these reforms wereparticularly effective for African American students.Maintaining and sustaining reform efforts at this school remain achallenge. Restructuring and recapturing processes may be expensive,time consuming, and dependent on dedicated, intelligent teachers andadministrators. The changes represent a dramatic shift away fromconventional mathematics education at the K-4 level. He's been [principal] hiring people who are wonderful teachers and wonderful people. But, he really, I guess he lets them know that a lot is expected of them. So when he is looking for a teacher, he's looking for someone who is going to come to school and not only follow his or her contract, but also do a lot more. I've noticed the teachers that he hired in the last couple of years; those are the ones who are here on Saturdays. If he's able to hire people that are totally dedicated like that and continue to hire teachers like this as people retire, the program will continue (fourth grade teacher).Maintaining theses reforms within a changing and controllingbureaucratic bu��reau��crat?n.1. An official of a bureaucracy.2. An official who is rigidly devoted to the details of administrative procedure.bu institution may be difficult. For example, some experiencedstaff members retire. Other experienced staff members leave for familyobligations and career advancements. It is not easy to acquire andeducate new administrators and new teachers about mathematics, socialconstructivist learning theory, and social constructivist instruction.This type of instructional reform requires ongoing passion toward adultlearning, creative instructional design Instructional design is the practice of arranging media (communication technology) and content to help learners and teachers transfer knowledge most effectively. The process consists broadly of determining the current state of learner understanding, defining the end goal of , and more instructional time forsome students.In addition to the uncertainty that comes with personnel changes,this type of reform may be threatened by district decisions to purchasetextbooks and instructional guides that package content, predetermine pre��de��ter��mine?v. pre��de��ter��mined, pre��de��ter��min��ing, pre��de��ter��minesv.tr.1. To determine, decide, or establish in advance: instruction, and give teachers a "crutch crutch(kruch) a staff, ordinarily extending from the armpit to the ground, with a support for the hand and usually also for the arm or axilla; used to support the body in walking. crutchn. " that allows them toavoid active and creative instruction. The reforms at the school wereachieved without standard textbooks because teachers and administratorstrusted their own abilities for creating active, interactive, andcontextual mathematics instruction.Overall, the reforms represent a major commitment to teacher andadministrator learning, to teacher as designer of instruction, toteacher as learner, to teacher as instructional leader, and to theeducation of all children even through it must happen outside theconventional school schedule. These kinds of reforms may not besustained without money, time, and the ongoing commitment of teachersand administrators.DiscussionCreating learning opportunities for all children to make sense ofimportant mathematics concepts may require restructuring and reculturingschools. "If the board educational goals of increased access andachievement for all students are to be reached, it is essential thatpolicies are put in place and actions are undertaken to enhance theteaching and learning of mathematics in poor communities" (NCTM2001a, p. 2). Fitting social constructivist theory into traditionalschool schedules and school resources may produce frustration anddisappointment until new teaching/learning environments and new timeschedules are designed. Findings suggest that social constructivisttheory helps teachers and learners move beyond the limited mathematicalinformation gained through practices designed from behaviorist theory.Kumon program is a behaviorist approach. It builds an importantfoundation for some students. It perfects computational Having to do with calculations. Something that is "highly computational" requires a large number of calculations. skills for somestudents. Nevertheless, acquiring computational skills withoutunderstanding and application seriously limits students' success inmathematics. Constructivist teaching is required for understanding andapplying mathematics.Reaching all students with mathematical understanding may alsorequire more money, more time, more resources and a radicalphilosophical shift for mathematics teachers and educational leaders. Weclose our discussion with a remark from the principal.There are so many unknowns. The structure [of existing schoolsystem] is not big enough for this reform. It doesn't fit.I've used comparisons like a box. The current instructional boxisn't big enough to hold this change. It's not just'thinking out-of-the box' either. That might be easier.It's that you don't have a box big enough to fit what youneed. The instructional frame isn't big enough to fit the art ofit, the beauty of it. You need a bigger frame. I've looked forappropriate metaphors or similes that would fit all of this and explainit to people clearly. It's not easy to explain this whole effort(principal).ReferencesBauersfeld, H. (1988). Interaction, construction, and knowledge:Alternative perspective for mathematics education. In T. Cooney & D.Grouws (Eds.). Effective mathematics teaching. (pp. 27-46). Reston, VA:NCTM.Bigelow, D., Harvey, B., Karp, S., & Miller, L. (Eds.). (2001).Rethinking our classrooms, Volume 2: Teaching for equity and justice.Milwaukee, WI: Rethinking Schools, Ltd.Burns, M. (1992). About teaching mathematics, a K-8 resource. MathSolutions Publications. Sausalito, CA.Campbell, P. F. (1996). Empowering children and teachers in theelementary mathematics classrooms of urban schools. Urban Education, 30,449-475.Campbell, P. F. (2001). When the vision confronts reality:Implementing reform in elementary school mathematics in an urban schooldistrict. Challenges in the mathematics education of African Americanchildren, Proceedings of the Benjamin Banneker This article requires authentication or verification by an expert.Please assist in recruiting an expert or [ improve this article] yourself. See the talk page for details. Association LeadershipConference. (pp. 45-50). Reston, VA: NCTM.Cobb, P. Wood, T., & Yackel, E. (1991). A constructivistapproach to second grade mathematics. In Von Lagerfeld (Ed.), RadicalConstructivism in Mathematics Education. (pp. 157-176). Dordrecht, theNetherlands: Kluwer Academics Press.Cobb, P. & Yackel, E. (1996). Constructivist, emergent emergent/emer��gent/ (e-mer��jent)1. coming out from a cavity or other part.2. pertaining to an emergency.emergent1. coming out from a cavity or other part.2. coming on suddenly. , andsociocultural so��ci��o��cul��tur��al?adj.Of or involving both social and cultural factors.soci��o��cul perspectives in the context of developmental research.Educational Psychology, 31, 175-190.Fennema, E., Franke, M. Carpenter, T., & Carey, D. (1993).Using Children's mathematical knowledge in instruction. AmericanEducational Research Journal, 30, 555-584.Frankenstein, M. (1995). Equity in mathematics education: Class inthe world outside the class. In W. G. Secada, E. Fennema, & L. B.Adajian (Eds.), New directions for equity in mathematics education. (pp.165-190). 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(pp. 131-162).Hillsdale, NJ: Lawrence Erlbaum Associations.Roland Pourdavood, Cleveland State University Cleveland State University,at Cleveland, Ohio; coeducational; founded 1964, incorporating Fenn College (est. 1923). The Cleveland-Marshall School of law was incorporated in 1969. Lawrence V. Svec, Lomond Elementary SchoolLynn M. Cowen, Onaway Elementary School**An earlier draft of this paper was presented at the Psychology ofMathematics Education (PME-NA PME-NA North American Chapter of the International Group for the Psychology of Mathematics Education ) in Athens Georgia Georgia, country, AsiaGeorgia(jôr`jə), Georgian Sakartvelo, Rus. Gruziya, officially Republic of Georgia, republic (2005 est. pop. 4,677,000), c.26,900 sq mi (69,700 sq km), in W Transcaucasia. , 2002.