자연주의적 탐구조사(Naturalistic Inquiry)

Lincoln, Yvonna, and Egon Guba, "Postpositivism and the Naturalist Paradigm," pp. 14-46 in Yvonna Lincoln and Egon Guba, Naturalistic Inquiry . Beverly Hills, CA: Sage, 1985.*

Pedagosko i psiholosko testiranje i procjenjivanje spadaju među najvažnije doprinose znanosti o ponasanju nasem drustvu i pružaju temeljna i znacajna poboljsanja u odnosu na ranije postupke. Iako se ne može ustvrditi da su svi testovi dovoljno usavrseni niti da su sva testiranja razborita i korisna, postoji velika kolicina informacija koje ukazuju na ucinkovitost kvalitetnih instrumenata u onim situacijama u kojima je njihovo koristenje potkrijepljeno validacijskim podacima. Pravilna upotreba testova može dovesti do boljih odluka o pojedincima i programima nego sto bi to bio slucaj bez njihovog koristenja, a također i ukazati na put za siri i pravedniji pristup obrazovanju i zaposljavanju. Međutim, losa upotreba testova može dovesti do zamjetne stete nanesene ispitanicima i drugim sudionicima u procesu donosenja odluka na temelju testovnih podataka. Cilj Standarda je promoviranje kvalitetne i eticne upotrebe testova te uspostavljanje osnovice za ocjenu kvalitete postupaka testiranja. Svrha objavljivanja Standarda je uspostavljanje kriterija za evaluaciju testova, provedbe testiranja i posljedica upotrebe testova. Iako bi evaluacija prikladnosti testa ili njegove primjene trebala ovisiti prvenstveno o strucnim misljenjima, Standardi pružaju okvir koji osigurava obuhvacanje svih relevantnih pitanja. Bilo bi poželjno da svi autori, sponzori, nakladnici i korisnici profesionalnih testova usvoje Standarde te da poticu druge da ih također prihvate.

Standards for educational and psychological testing

미국의 “전국 수학 교사 협의회”(National Council of Teachers of Mathematics, NCTM)는 1989년부터 〈학교 수학의 교육과정과 평가 규준〉(1989), 〈수학 가르침(교수)의 전문성 규준〉(1991), 〈학교 수학의 평가(시험) 규준〉(NCTM, 1995), 〈학교 수학의 원리와 규준〉(2000)을 출판하여 미국의 수학 교육 의 전망(목표, 나아갈 길)과 규준(실행 지침)을 제시하였다. 수학 교사들로 구성된 미국의 NCTM은 학생, 학부모, 학교 행정가 등 많은 사람들과 힘을 합하여 모든 학생들에게 수준 높은 수학 교육을 받을 수 있는 여건(환경, 기회)을 조성하는 데 구심점의 역할을 하였다. 한편 많은 관련 단체들은 여러 배경과 능력을 가진 학생들이 전문성을 지닌 교사(특수 교사를 일컫는 밀이 아니다. 수학 교과를 이해하고 수학의 전문성과 특수성을 가르칠 수 있는 일반 교사를 일컫는 말이다.)로부터 미래를 대비해 평등하고, 진취적이며, 지원이 잘 이루어지고, 공학 도구(IT)가 잘 갖춰진 환경에서 중요한 수학적 아이디어를 이해하면서 학습할 수 있는 수학 교실(미국에서는 우리나라처럼 수학 교사가 수학 시간에 학생의 방(교실: Homeroom)에 찾아가지 않고 학생들이 선생의 방(수학 교실: Classroom)을 찾아온다. 전형적인 수학 교실의 사진은 2쪽에 나와 있다.)을 만들기 위해 함께 힘썼다. NCTM에서 출간한 여러 규준들은 우리나라의 제6차와 제7차 교육과정에도 큰 영향을 미쳤다. 이 글에서는 NCTM(2000)에서 제시한 학습 원리를 간단히 살펴본 다음 이를 중심으로 현재 미국 수학 교육의 교수ㆍ학습 이론의 동향을 살펴본다.

미국 NCTM의 Principles and Standards for School Mathematics에 나타난 수학과 교수,학습의 이론

What Video Games Have to Teach Us about Learning and Literacy

Video has assumed an increasingly prominent role in teacher education, particularly in the form of the viewing of videotaped class lessons by preservice teachers. Yet there is little research that confirms whether preservice teachers attend to the aspects of the video(s) that teacher educators anticipate or desire. This article explores this issue and reports on the impact of video viewing as a means to improve teachers’ ability to be observers of classroom practice. We utilized a pre- and post-test design to measure the quantity and type of classroom events that preservice mathematics teachers noticed before and after a teaching methods course where improving observation skills was an explicit goal. The results of the pre-assessment suggest that preservice teachers generally do not enter teaching methods courses with well-developed observation skills. The post-assessment indicates that the course led to significant increases in preservice teachers’ observation skills, particularly in teachers’ ability to notice features of the classroom environment, mathematical content of a lesson, and teacher and student communication during a lesson.

Learning to observe: using video to improve preservice mathematics teachers’ ability to notice

Encouraging students to share and compare solution methods is a key component of reform efforts in mathematics, and comparison is emerging as a fundamental learning mechanism. To experimentally evaluate the effects of comparison for mathematics learning, the authors randomly assigned 70 seventhgrade students to learn about algebra equation solving by either (a) comparing and contrasting alternative solution methods or (b) reflecting on the same solution methods one at a time. At posttest, students in the compare group had made greater gains in procedural knowledge and flexibility and comparable gains in conceptual knowledge. These findings suggest potential mechanisms behind the benefits of comparing contrasting solutions and ways to support effective comparison in the classroom.

Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations.

Reconceptualizing procedural knowledge.

Manipulatives are usually used in lower grades in the initial stages of learning as teachers introduce basic concepts with whole numbers. This exposure to concrete objects is often fleeting and transitory. The use of manipulatives in upper elementary school grades is virtually nonexistent.117 117. Howard, Perry, and Lindsay (1996); Howard, Perry, and Conroy (1995). RECOMMEnDATIOn 5. InTERvEnTIOn MATERIAlS ShOulD InCluDE OPPORTunITIES fOR ThE STuDEnT ( 33 ) The panel suggests that the interventionist use concrete objects in two ways. First, in lower elementary grades, use concrete objects more extensively in the initial stages of learning to reinforce the understanding of basic concepts and operations.118 Concrete models are routinely used to teach basic foundational concepts such as place value.119 They are also useful in teaching other aspects of mathematics such as multiplication facts. When a multiplication fact is memorized by question and answer alone, a student may believe that numbers are to be memorized rather than understood. For example, 4 × 6 equals 24. When shown using manipulatives (as in example 7, p. 35), 4 × 6 means 4 groups of 6, which total as 24 objects. 118. Darch (1989); Fuchs, Seethaler et al. (2008); Fuchs, Fuchs, Craddock et al. (2008). 119. Fuchs et al. (2005); Fuchs, Seethaler et al. (2008); Fuchs, Powell et al. (2008). Second, in the upper grades, use concrete objects when visual representations do not seem sufficient in helping students understand mathematics at the more abstract level. Use manipulatives expeditiously, and focus on fading them away systematically to reach the abstract level.120 In other words, explicitly teach students the concepts and operations when students are at the concrete level and consistently repeat the instructional procedures at the visual and abstract levels. Using consistent language across representational systems (manipulatives, visual representations, and abstract symbols) has been an important component in several research studies.121 Example 8 (p. 35) shows a set of matched concrete, visual, and abstract representations of a concept involving solving singlevariable equations. 120. Fuchs et al. (2005); Witzel (2005); Witzel, Mercer, and Miller (2003). 121. Fuchs et al. (2005); Butler et al. (2003); Witzel (2005); Witzel, Mercer, and Miller (2003). Example 4. Representation of the counting on strategy using a number line 0 1 2 3 4 5 6 7 8 9 10 RECOMMEnDATIOn 5. InTERvEnTIOn MATERIAlS ShOulD InCluDE OPPORTunITIES fOR ThE STuDEnT ( 34 ) Example 5. Using visual representations for multidigit addition Example 6. Strip diagrams can help students make sense of fractions A group of ten can be drawn with a long line to indicate that ten ones are joined to form one ten: Simple drawings help make sense of two-digit addition with regrouping: 36

/pdf/assisting-students-struggling-with-mathematics-response-to-4j91w5z179.pdf

Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools. NCEE 2009-4060.

Researchers in both cognitive science and mathematics education emphasize the importance of comparison for learning and transfer. However, surprisingly little is known about the advantages and disadvantages of what types of things are being compared. In this experimental study, 162 seventh- and eighth-grade students learned to solve equations (a) by comparing equivalent problems solved with the same solution method, (b) by comparing different problem types solved with the same solution method, or (c) by comparing different solution methods to the same problem. Students' conceptual knowledge and procedural flexibility were best supported by comparing solution methods and to a lesser extent by comparing problem types. The benefits of comparison are augmented when examples differ on relevant features, and contrasting methods may be particularly useful in mathematics learning.

/pdf/compared-with-what-the-effects-of-different-comparisons-on-3thf7ibzgk.pdf

Jon R. Star

Papers

Learning to observe: using video to improve preservice mathematics teachers’ ability to notice

Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations.

Reconceptualizing procedural knowledge.

Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools. NCEE 2009-4060.

Compared with What? The Effects of Different Comparisons on Conceptual Knowledge and Procedural Flexibility for Equation Solving.