Bethany Rittle-Johnson

Bio: Bethany Rittle-Johnson is an academic researcher from Vanderbilt University. The author has contributed to research in topics: Procedural knowledge & Concept learning. The author has an hindex of 38, co-authored 114 publications receiving 6277 citations. Previous affiliations of Bethany Rittle-Johnson include Carnegie Mellon University.

Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process.

Abstract: The authors propose that conceptual and procedural knowledge develop in an iterative fashion and that improved problem representation is 1 mechanism underlying the relations between them. Two experiments were conducted with 5th- and 6th-grade students learning about decimal fractions. In Experiment 1, children's initial conceptual knowledge predicted gains in procedural knowledge, and gains in procedural knowledge predicted improvements in conceptual knowledge. Correct problem representations mediated the relation between initial conceptual knowledge and improved procedural knowledge. In Experiment 2, amount of support for correct problem representation was experimentally manipulated, and the manipulations led to gains in procedural knowledge. Thus, conceptual and procedural knowledge develop iteratively, and improved problem representation is 1 mechanism in this process.

Conceptual and procedural knowledge of mathematics: Does one lead to the other?

Abstract: ages understand and what they struggle to learn, and examine how instruction influences children's acquisition of both concepts and procedures. The purpose of the present study was to explore the relations between conceptual and procedural knowledge in children learning the principle that the two sides of an equation represent the same quantity. Specifically, the study investigated how instruction about the concept of mathematical equivalence influences children's problem-solving procedures and how instruction about a problem-solving procedure influences children's conceptual understanding of equivalence. In addressing these issues, we also identified what aspects of equivalence fourth- and fifth-grade students understand, what aspects they do not understand but can easily learn, and what aspects they have difficulty learning.

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

Abstract: 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.

Promoting Transfer: Effects of Self-Explanation and Direct Instruction

Abstract: Explaining new ideas to oneself can promote transfer, but how and when such self-explanation is effective is unclear. This study evaluated whether self-explanation leads to lasting improvements in transfer success and whether it is more effective in combination with direct instruction or invention. Third- through fifth-grade children (ages 8-11; n=85) learned about mathematical equivalence under one of four conditions varying in (a) instruction on versus invention of a procedure and (b) self-explanation versus no explanation. Both self-explanation and instruction helped children learn and remember a correct procedure, and self-explanation promoted transfer regardless of instructional condition. Neither manipulation promoted greater improvements on an independent measure of conceptual knowledge. Microgenetic analyses provided insights into potential mechanisms underlying these effects.

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

Abstract: 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.

Standards for educational and psychological testing

Abstract: 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.

Adding It Up: Helping Children Learn Mathematics

Abstract: Adding It Up explores how students in pre-K through 8th grade learn mathematics and recommends how teaching, curricula, and teacher education should change to improve mathematics learning during these critical years. The committee identifies five interdependent components of mathematical proficiency and describes how students develop this proficiency. With examples and illustrations, the book presents a portrait of mathematics learning: * Research findings on what children know about numbers by the time they arrive in pre-K and the implications for mathematics instruction. * Details on the processes by which students acquire mathematical proficiency with whole numbers, rational numbers, and integers, as well as beginning algebra, geometry, measurement, and probability and statistics. The committee discusses what is known from research about teaching for mathematics proficiency, focusing on the interactions between teachers and students around educational materials and how teachers develop proficiency in teaching mathematics.

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

Abstract: 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.