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2012-11-27 Clements & Sarama, 2011, Early childhood teacher education- The case of geometry

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2012-11-27 Clements & Sarama, 2011, Early childhood teacher education- The case of geometry
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  Early childhood teacher education: the case of geometry Douglas H. Clements  • Julie Sarama Published online: 23 February 2011   Springer Science+Business Media B.V. 2011 Abstract  For early childhood, the domain of geometry and spatial reasoning is animportant area of mathematics learning. Unfortunately, geometry and spatial thinking areoften ignored or minimized in early education. We build a case for the importance of geometry and spatial thinking, review research on professional development for theseteachers, and describe a series of research and development projects based on this body of knowledge. We conclude that research-based models hold the potential to make a sig-nificant difference in the learning of young children by catalyzing substantive change in theknowledge and beliefs of their teachers. Keywords  Scaling up professional development    Geometry    Spatial reasoning   Learning trajectories    Early childhood Introduction Early childhood teacher education: the case of geometryFor early childhood, the domain of geometry and spatial reasoning is an important areaof mathematics learning (NCTM 1991, 2006). Viewed broadly, geometric and spatial thinking are not only important in and of themselves, but they also support number andarithmetic concepts and skills (Arcavi 2003). Research even suggests that the ability torepresent  magnitude  is dependent on visual-spatial systems in regions of the parietal cortexof the brain (Geary 2007; Pinel et al. 2004; Zorzi et al. 2002). Unfortunately, geometry and spatial thinking are often ignored or minimized in both early education (Sarama andClements 2009) and in the professional development of early childhood teachers(H. P. Ginsburg et al. 2006). In this article, we build a case for the importance of geometry D. H. Clements ( & )    J. SaramaGraduate School of Education, University at Buffalo, State University of New York,212 Baldy Hall (North Campus), Buffalo, NY 14260-1000, USAe-mail: clements@buffalo.edu  1 3 J Math Teacher Educ (2011) 14:133–148DOI 10.1007/s10857-011-9173-0  and spatial thinking, review research on professional development for these teachers, anddescribe a series of research and development projects based on this body of knowledge.The importance of geometry and spatial thinking for young studentsSome mathematicians have claimed that, except for simple calculation, geometric conceptsunderlie all of the mathematical thought (e.g., Bronowski 1947). Smith (1964) argued that mathematics is a special kind of language through which we communicate ideas that areessentially spatial. From number lines to arrays, even quantitative, numerical, and arith-metical ideas rest on a geometric base. Cross-cultural research substantiates that coregeometrical knowledge, like implicit basic number or quantitative knowledge, appears tobe a universal capability of the human mind (Dehaene et al. 2006).Geometry can serve as a core-relating science and mathematics. Two of the mostprominent physicists of the last 100 years attributed their advancements to geometry. As aboy, Einstein was fascinated with a compass, leading him to think about geometry andmathematics. He taught himself extensively about geometry by age 12. Later in life,Einstein said that his elements of thought were always initially of a geometric and spatialnature, including ‘‘certain … more or less clear images which can be voluntarily reproducedor combined. Conventional words or other signs have to be sought for laboriously only in asecondary stage, when the associative play is sufficiently established.’’ Hawking put it thisway: ‘‘Equations are just the boring part of mathematics. I attempt to see things in terms of geometry’’ (Larsen 2005, p. 43). These two are not alone, visual thinking played a dom-inant role in the thinking of Michael Faraday, Sir Fancis Galton, Nikoa Tesla, JamesD. Watson, Rene´ Thom, and Buckminster Fuller, among others (Shepard 1978). Thisperspective is summarized by a modern geometer as follows:Geometry should be a focus at every age, in every grade, every year. Mathematicscurricula are often criticized for their insularity—‘what does this have to do with thereal world?’ No mathematical subject is more relevant than geometry. It lies at theheart of physics, chemistry, biology, geology and geography, art and architecture. Italso lies at the heart of mathematics, though through much of the twentieth centurythe centrality of geometry was obscured by fashionable abstraction. This is changingnow, thanks to computation and computer graphics which make it possible to reclaimthis core without loss of rigor. The elementary school curriculum should give thechildren the tools they will need tomorrow. (Marjorie Senechal, personal commu-nication, November 18, 2005)Spatial thinking is an essential human ability that contributes to mathematical ability. Itis a process that is distinct from verbal reasoning (Shepard and Cooper 1982) and functionsin distinct areas of the brain (Newcombe and Huttenlocher 2000). Further, mathematicsachievement is related to spatial abilities (e.g., Ansari et al. 2003). As an example,empirical evidence indicates that spatial imagery reflects not just general intelligence butalso a specific ability that is highly related to ability to solve mathematical problems,especially nonroutine problems (e.g., Wheatley et al. 1994). This is particularly importantbecause some individuals are harmed in their progression in mathematics due to lack of attention to spatial skills, benefit from more geometry and spatial skills education(e.g., Casey and Erkut 2005).Education in geometry may contribute to a growth in mathematical competence and inother cognitive abilities, including IQ (Clements and Battista 1992; Clements and Sarama2007b). Geometric knowledge, in particular, is highly related to mathematical reasoning 134 D. H. Clements, J. Sarama  1 3  and a host of other mathematics concepts and skills (Tatsuoka et al. 2004), includingproportional reasoning, judgmental application of knowledge, concepts and properties, andmanaging data and processing skills, leading the authors to conjecture that geometry maybe a gateway skill to the teaching of higher-order mathematics thinking skills.Considering these bodies of research and professional judgments, students may benefitfrom attention to geometric and spatial thinking from the earliest years. Less salient, butperhaps just as important, are the competencies of students’ learning of related topics inother subject-matter domains such as computer graphics, navigation, geography, visualarts, and architecture. The U.S. Employment Service estimates that most technical-sci-entific occupations such as drafter, airplane designer, architect, chemist, engineer, physi-cist, and mathematician require persons having spatial ability at or above the 90thpercentile.International studies indicate a weakness in students’ geometric achievements (Mulliset al. 1997). In the Third International Mathematics and Science Study (TIMSS) research,international performance on geometry and measurement is low, especially for somecountries (Beaton et al. 1996; Ginsburg et al. 2005; Lappan 1999). These deficits have been identified in the earliest years of life. Improved teacher education is needed to addresslimited student learning of geometry (with some empirical support for this causalconnection, e.g., van der Sandt 2007). Professional development Geometry and present-day preparation of teachersAlthough there are exceptions, teachers in many countries, including the U.K. (Jones 2000)and South Africa (van der Sandt 2007), and [throughout the pre-K to grade 12], are notalways provided with adequate preparation in geometry and the teaching and learning of geometry. Of all mathematics topics, geometry was the one prospective teachers claimed tohave learned the least and believed they were least prepared to teach (Jones et al. 2002).We shall provide several illustrations of this lack of knowledge, with a caveat: thisresearch base is small and limited in focus. For this reason, the illustrations do not reflectthe broad range of concepts, skills, and perspectives that define what we believe to beimportant in the domains of geometry and spatial thinking. However, they do suggestteachers’ unfamiliarity with those domains. For example, many prospective teachers onlyreach level 1 of the van Hiele (1989) model of geometric thinking—recognizing andcategorizing shapes only on the basis of their overall physical similarity to prototypes (‘‘itmust be a rectangle, because it looks like a door’’). Many are not guided to reach level 2,the descriptive/analytic level, at which people recognize and characterize shapes by theirproperties (Clements 2003; Swafford et al. 1997). In one study, for example, prospective teachers in the U.K. were at best thinking at level 2 (Fujita and Jones 2006a).This lack of attention to geometry reveals itself in prospective teachers’ responses totasks in this study (Fujita and Jones 2006a). For example, approximately 13% of pro-spective teachers in Scotland identified a square as a rectangle and approximately 18%realized that a parallelogram is a trapezium (the authors note that such results contrastswith Kawasaki’s (1992) findings that 73% of Japanese prospective teachers define a tra-pezium correctly, as cited in Fujita and Jones 2006b). Although almost all prospectiveteachers could draw a square, almost 2/3 could not define it correctly, leaving out anymention of angles (or other constraining properties). Probably, the geometry-deprived Early professional development: geometry 135  1 3  (Fuys et al. 1988) elementary and middle school education that these prospective teachersexperienced ‘‘fixed’’ their image and definition of a square, and they see no need tomention angles, even when it is necessary. For all ages, people who know a correct verbaldescription of a concept but possess a limited visual image (or concept image) associatedstrongly with the concept may have difficulty learning and applying the descriptions. Theyare influenced most by the visual image (see the theory of hierarchic interactionalism inSarama and Clements 2009, and the construct of concept images—a combination of all themental pictures and properties that have been associated with the concept in a person’smemory). In another study, 70% of prospective elementary teachers were below van Hielelevel 3, at which people understand relationships between  classes  of figures. Several wereat the pre-recognition level (level 0) (Sarama and Clements 2009). Almost two-thirds wereat the visual level (1).Although data on teachers of the youngest children is rare, our research with hundredsof pre-school teachers (discussed in a succeeding section) suggests that most earlychildhood teachers also have not attained adequate levels of geometric knowledge. This isconsistent with the finding that teachers of young children are provided with very limitedprofessional development in mathematics (H. P. Ginsburg et al. 2006).Such limited education, from their own early years on, leaves teachers under-preparedfor teaching geometry. Although level 3 relational thinking in geometry and spatial sense(relations between classes of shapes) is not necessarily a goal of early childhood education,there are several reasons that teachers’ low level of geometric thinking is a concern. First,teachers at the pre-recognition (0) and even visual (1) levels cannot adequately assess andteach children at any level, as such thinking is not consistent with mathematical properties(effective teachers must know considerably more than simply the content of the age/gradethey teach, National Mathematics Advisory Panel 2008). Second, even if goals for childrendo not immediately address level 3 thinking, we wish to  do no harm.  Teachers who ask children to go on a ‘‘shape hunt’’ for rectangles, and reject a child’s choice because shefound a square, make it necessary to carry out later on the difficult task of ‘‘unteaching.’’Third, as stated previously, such limits in levels of thinking reliably reflect a general lack of awareness of the broad domains of geometry and spatial thinking. Thus, there is a needfor substantial professional development for teachers of young children.A lack of knowledge of geometry and geometry education affects new generations. Forexample, an early study found that kindergarten children had a great deal of knowledgeabout shapes and matching shapes before instruction began. In one episode, the teachertended to elicit and verify children’s prior knowledge but did not seem to add or developnew knowledge. That is, about two-thirds of the interactions between the teacher andchildren had children repeat what they already knew in a repetitious format as in thefollowing exchange: Teacher: Could you tell us what type of shape that is? Children: Asquare. Teacher: Okay. It’s a square (Thomas 1982). Along with the researcher’s pre-test,this illustrates the common finding that children may already possess the information theteacher is trying to present. When teachers did elaborate, their statements were often filledwith mathematical inaccuracies. For instance, teachers claimed that all ‘‘diamonds’’(rhombi) are squares, that two triangles put together always make a square, and that asquare cut in half always yields two triangles.A mathematics survey was sent to 3,000 teachers and administrators from day care,family care, traditional nursery schools, Head Start, and public and parochial pre-schools intwo states in the United States, of which over 400 complete surveys were returned. Askedabout their main mathematics activities, 67% of early childhood care providers chosecounting; 60% chose sorting; 51%, numeral recognition; 46%, patterning; 34%, number 136 D. H. Clements, J. Sarama  1 3  concepts; 32%, spatial relations; 16%, making shapes; and 14%, measuring (Sarama 2002;Sarama and DiBiase 2004). Geometry and measurement concepts were the least popular.This is another reason professional development is so important for teachers for youngchildren.Strategies for effective professional developmentThe single most dominant factor affecting students’ academic progress is the effectivenessof their teachers (Wright et al. 1997). The presence of   cumulative  effects of teachers onstudent achievement—with little evidence of compensatory effects of more effectiveteachers in later grades (Sanders and Rivers 1996)—indicates that teachers of the earlyyears must be effective to serve children well, especially those at the lower end of enteringknowledge. Unfortunately, as we have seen, there is little in the literature that wouldindicate that there are many effective early childhood teachers of geometry. Therefore,professional development for these teachers is critical.Good professional development in geometry may not be easy to conduct. Some pro-fessional development programs do nothing to increase geometric knowledge (Fujita andJones 2006b; Jones 2000), and teachers’ knowledge can even degrade during such pro- grams (van der Sandt and Nieuwoudt 2004). It appears that hearing, reading, and mem-orizing formal definitions makes little or no impact on these prospective teachers. Even insome intervention projects with an extensive professional development component,teachers improve only in their teaching of number tasks to young children—the children’sgeometry and geometry reasoning were not affected (Campbell and Rowan 1995).Geometry knowledge may have been inadequately addressed. Still there are indicationsthat some professional development can make a difference to both teachers and theirstudents (Jacobson and Lehrer 2000; Swafford et al. 1997). The difference may lie in the nature of the professional development. General require-ments and measures such as certification alone are not reliable predictors of high-qualityteaching (Early et al. 2007; National Mathematics Advisory Panel 2008). More direct mea- sures of what the teachers know about mathematics and the learning and teaching of math-ematicsdopredictthequalityoftheirteaching(NationalMathematicsAdvisoryPanel2008).For example, the mathematics achievement gains of 1st and 3rd graders were significantlyrelated to their teachers’ mathematical knowledge for teaching (Hill et al. 2005).Professional development ought to focus on geometry and spatial reasoning and be suf-ficiently intense and extensive. As an example, one 4-week (3 h/day, 4 days per week)intervention program designed to enhance teachers’ knowledge of geometry and theirknowledge of research on student cognition in geometry resulted in significant positivechanges in content knowledge and van Hiele level for middle school teachers, as well as inwhattheytaughtandhowtheytaught(Swaffordetal.1997).Initially,participantsfunctionedat low van Hiele levels, with 79% being at the first three levels, 75% were at the top twolevels after professional development (formal deduction and rigor). Progression to a highervan Hiele level may be rapid if facilitated by high-quality instruction for adult learners. Developing and scaling up effective professional development for early childhoodgeometry Our research for the last decade has addressed how to design and implement high-qualityprofessional development that improves the competencies of the participating teachers and, Early professional development: geometry 137  1 3
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