## ANNOTATED BIBLIOGRAPHY -- CLASS PROJECT

Aiken, L.R. (1970). Attitude towards mathematics.

Review of Educational Research,40(4), 551-596.There has been a dramatic increase in the study of attitudes towards mathematics since 1960. This article is a review of the literature on the types of factors that influence one’s attitude towards mathematics. The first part of the paper deals with the methods of measuring attitudes towards mathematics. The rest of the article investigates the factors that affect attitude and the relationship it has on a student’s performance. Aiken found that attitudes towards mathematics can be traced back to childhood, and attitudes tend to become more negative as students reach higher grades in school. The research also suggests that “attitudes affect achievement and achievement in turn affects attitudes” (Aiken, 1970). Aiken examines the differences between attitudes among different grade levels: elementary, middle and high school). The article continues to look at the relationship of attitudes to personality and social factors and also focuses on teacher characteristics, attitudes, behavior, instructional methods and curriculum. Aiken concludes with research on developing positive attitudes and modifying negative attitudes. (C. Foy).

Asante, K. O. (2012). Secondary students' attitudes towards mathematics.

IFE PsychologIA20(1), 121-133.This article describes a research study that investigated high school students’ attitudes towards mathematics and gender differences in attitudes towards mathematics in Ghana. The study used two questionnaires that examined demographic information and the Attitudes Towards Mathematics Inventory. The results showed a significant difference in attitudes towards mathematics between boys and girls. This agrees with most of the previous research in the subject area. Most of the studies reported that girls lack confidence in mathematics compared to boys. The girls tended to view math as a male domain and were anxious about mathematics as well. The causes of the gender differences in mathematics attitude may be different in Ghana than the rest of the world. However, the results were still similar. The major role of gender differences towards mathematics in Ghana seemed to be a result of socialization into varied gender roles. (C. Foy)

Aspinwall L., & Miller, L. D. (2001). Diagnosing conflict factors in calculus through students’ writings. One teachers’ reflections.

Journal of Mathematical Behavior20(1), 89-107. Retrieved from http://www.sciencedirect.com/science/article/pii/S0732312301000633#

In order to inform teachers about students' understanding of derivatives and integrals in a calculus course, Aspinwall and Miller study the results of a teacher-as-researcher incorporating and analyzing writing prompts in his curriculum. The writing prompts "were intended to diagnose any existing conflict between a student's concept image and concept definition" based on the definitions presented by Tall and Vinner (p. 92). The course aimed to present the material in three ways: numerically, graphically, and analytically. However, the student responses revealed that while some seemed to show that their concept image retained features from the formal concept definition presented in class, others wrote responses that the teacher felt required follow-up questions to clarify, expand, or correct the students’ responses. For instance, some students only used formulas while others reversed rolls for lower and upper bounds for an integral. The authors encourage using carefully written writing prompts in calculus to help diagnose and resolve conflict factors, and present students with opportunities to “practice communicating their mathematical understanding” (p. 106). (I. Stevens)

Ball, D. L., & McDiarmid G. W. (1989). The Subject Matter Preparation of Teachers. National Center for Research on Teacher Education, East Lansing, MI, May 1989, Issue Paper 89(4), 1-29.

“Although subject matter knowledge is widely acknowledged as a central component of what

teachers need to know, research on teacher education has not, in the main, focused on the

development of teachers' subject matter knowledge”. This paper focuses on the subject matter preparation of teachers: what subject matter preparation entails, where and when it occurs, and with what outcomes. Ball and McDiarmid further propose a framework that can contribute to future research in this area. The paper is organized into three parts: a) examine the concept of subject matter knowledge, b) offers a framework for the sources and outcomes of teachers’ subject matter learning, c) use the framework to consider extant evidence about teachers’ subject matter preparation. Authors stress that understanding of subject matter come from both in and outside the classroom. More research needs to be done to streamline the in and outside classroom experiences of future teachers to enhance their knowledge in subject matter. (A. Kar)

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education.

Elementary School Journal, 90(4),449–466.http://www.jstor.org/stable/1001941Deborah Ball asserts that mathematical understanding should be a primary concern for future math teachers, and not ignored in favor of content knowledge and teacher methodology. A selection of secondary and elementary pre-service teachers are evaluated based on how effectively they can create or identify real world example for a fraction division problem, and on their approach to explaining mathematical concepts. Most students were unsuccessful at coming up with matching examples. If a student could come up with an explanation for a mathematical concept, then it would usually consist of reciting mathematical rules. When interviewed about the nature of mathematics, most students described it as a set of rules and procedures. Ball argues that this demonstrates the interconnection between a student’s attitude about math and their understanding of it and demonstrates that pre-service teachers need more emphasis in this area. (J. Traxler)

Bartell, T. G. (2011). Learning to Teach Mathematics for Social Justice: Negotiating Social Justice and Mathematical Goals.

Journal for Research in Mathematics Education,44(1), 129-163. http://www.nctm.org/publications/article.aspx?id=35215

The article talks about how to learn and implement techniques to teach mathematics for social justice. There has been a lot of research done in inequity and social justice in mathematics classroom. But there have been few research that actually teaches the preservice teachers how to implement social justice in their classroom. Bartell stress that more work needs to be done for preservice teachers that increases understanding and expand their assumptions about the goals of mathematics education and the knowledge required of teachers negotiating various aspects of such practice. Bartell works with 8 secondary mathematics teacher in a graduate level mathematics course. Those students were divided into 2 groups. These two groups went from defining mathematics for social justice to development of lesson plans and analyzing the results of their lessons. Later Bartell summarizes her findings and suggests improvement in this area. (A. Kar)

Note: AJRME Equity Special Issuewas constructed by an appointed panel concerned with teaching mathematics for social justice. That panel included

Beatriz D'Ambrosio, Miami University

Danny Martin, University of Illinois at Chicago

Marilyn Frankenstein, University of Massachusetts, Boston

Judit Moschkovich, University of California, Santa Cruz

Rochelle Gutierrez, University of Illinois at Urbana-Champaign

Edd Taylor, Northwestern University

Signe Kastbert, Purdue University

David Barnes, NCTM

That special issue was made available online in 2011. In January 2013, Issue 1, Volume 44 of JRME published the special issue in printed form. The individual articles from 2011 can still be found with a Google search. Here is a link to theIntroductionto the special issue that was written by the Editorial Panel. It is a very interesting overview. Although, the claim is made that the print versity has the complete online materials, it is clear from just the abstract of Bartell's article that some editorial changes were made between 2011 and 2013. (J. Wilson)Begle, E. G. (1979)

Critical variables in mathematics education: Findings from a survey of the Empirical Literature. Washington, DC: MAA.Professor Begle acquired an immense library of mathematics education research produced up through the 1970s and undertook a synthesis of that research. His rationale for the review and this book are summed up in a position first stated at the ICME 1969 conference: "I see little hope for any further substantial improvements in mathematics education until we turn mathematics education into an empirical science, until we abandon our reliance on philosophicl discussions based on dubious assumptions, and instead follow a carefully constructed pattern of observation and speculation . . ." (pp. x-xi) The book is comprehensive and several chapters are relevant to issues of mathematics instruction. For example, Chapter 7 Instructional Variables discusses 22 different categories of variables EGB identifies as instruction. The book was a work in progress at the time of his death and was published essentially as he left it. (J. Wilson)

Brummelman, E., Thomaes, S., Overbeek, G., de Castro, B. O, van den Hout, M. A, & Bushman, B. J. (2013). On feeding those hungry for praise: Person praise backfires in children with low self-esteem.

Journal of Experimental Psychology: General, 1-6. Advance online publication. doi: 10.1037/a0031917

This article contains two research studies, evaluating 1) the usage and 2) impact of person and process praise on children’s self esteem. These authors argue, “certain forms of praise (in their belief, person praise) can backfire, especially on children with low self esteem” (p.1). The researchers first study included 357 Dutch parents and given scenarios, were ask what praise statement they would give a child. The researchers hypothesize that parents would give children with high self esteem more process praise and children with lo self esteem more person praise to protect them from feelings of shame and worthlessness, the very feelings parents inevitably want to protect their children from feeling. From the confirmed hypothesis in this study, their 2nd study including 313 students between the ages 8-13 aimed to evaluate the impacts of “how [person, process, and no] praise affects those who seem to need it the most—children with low self-esteem” (p.4). The researchers findings provide additional evidence on the notion that person praise has detrimental affects on student’s self-perception, especially students with low self-esteem. (C. Ramsey)

(This is a recently accepted article that has been published on line but is not yet in print.)

Eccles, J. S., & Jacobs, J. E. (1986). Social forces shape math attitudes and performance.

Signs: Journal of Women in Culture and Society,11(2), 367-380.This article discusses some major concerns in gender differences in mathematics education. It has been shown that males are more likely to score higher on standardized mathematics achievement tests than females, males are more likely than females to engage in mathematical extracurricular activities, and males typically perform better than females on spatial-visualization tests. Eccles and Jacobs mention a particular study by Benbow and Stanley in 1980, which found that boys scored higher than girls on the mathematics section of the SAT. Benbow and Stanley argued that this male dominance in mathematical ability is the best explanation for gender differences in mathematics. Eccles and Jacobs argue against this. They don’t believe that the SAT-M is an accurate measure of mathematical aptitude. In fact, Eccles and Jacobs argue that there were more factors in play that affected the difference in scores. Thus Eccles and Jacobs focus their study on other social and attitudinal factors that have a greater influence on mathematical achievement rather than just suspected innate ability. (C. Foy). (

A link has been provided to Benbow and Stanley (1980) since this annotation cites a disagreement between them and Eccles and Jacobs (1986). The Benbow and Stanley articles has been widely cited. See a Google search on Benbow and Stanley 1980. The PDF copy of this article has many spelling errors that must be carefully negotiated as you are reading.)

Bingolbali, E., & Monaghan, J. (2008). Concept image revisited.

Educational Studies in Mathematics,68(1), 19-35.This is a study done on mechanical engineering and mathematics undergraduates taking a Calculus course at a university in Turkey. While recognizing the work of Tall and Vinner (1981), the authors want to expand on the ideas of concept definition and concept image by considering how students’ departmental affiliation affects students’ concept images. The results of the study indicated that students from each of the departments initially showed no significant difference in their ideas of derivatives, but the post-test revealed that the mechanical engineering students performed significantly better on problems concerning rate of change and mathematics students performed significantly better on problem concerning tangent lines. From the post tests, the authors concluded that “students’ perceptions of their department impact upon how they position themselves and their developing preferences and concept images” (p. 31). (I. Stevens)

Blanton, M. L., Berenson, S. B., Norwood, K. S. (2001). Using classroom discourse to understand a prospective mathematics teacher’s developing practice.

Teaching and Teacher Education 17, 227-242.

This article centers on understanding the nature of classroom discourse and its role in Mary Ann (pseudonym), one mathematics student teacher’s developing practice. A Vygotskian approach of social interaction in learning is used to examine the nature of classroom discourse. Here the authors emphasized on verbal interaction in the classroom and how various routines often embedded in language, comprise the patterns of interaction in the classroom. Article suggests, “early classroom discourse of Mary Ann seemed to mediate Mary Ann’s pedagogy toward a more univocal paradigm of giving information and inspecting the accuracy of transmission” (p. 233). Later on Mary Ann experienced the power of students’ ideas through verbal interactions, her role changed significantly from being a teller to a participant. The article suggests that the classroom discourse helped mediate Mary Ann’s practice of teaching. (S. Ghosh Hajra)

Blanton, M. L., & Stylianou, D. A. (2011). Developing students’ capacity for constructing proofs through discourse.

The Mathematics Teacher(2), 140-145. http://www.jstor.org/stable/10.5951/mathteacher.105.2.0140., 105This article is about an observational study that was done in a mathematics classroom with a year long objective of developing better proof-writing and understanding skills among students. The class was taught by one of the authors and the classes were videotaped and then analyzed. The students in the classroom were given something to prove and then began their own basic attempts at the proof. The first proofs that the students gave were empirical based arguments, meaning they contained mainly numbers. The teacher would then use a classroom discussion to talk about the reasoning behind the students’ proofs. Since the original proofs used primarily numbers, the teacher orchestrated the discussion in such a way as to lead students towards a more deductive argument. By questioning and asking for clarification and explanations, the teacher made the students realize that they had to provide a better argument that could work for any arbitrary case. By working through discussions, the teacher allowed for the students to develop ideas and it promoted their understanding of definitions and theorems. Writing proofs is generally thought of as a difficult task in mathematics, but it is one of the most important parts. Proofs require a deep understanding of definitions and properties in mathematics. I think that if teachers just overlook the way that proofs are taught in the classroom, it can result in a huge lack of mathematical understanding among students. (C. Foy)

Bostic, J., & Jacobbe, T. (2010). Promote Problem-Solving Discourse.

Teaching Children Mathematics,17(1), 32-37.The authors, Jonathan Bostic and Tim Jacobbe, joined a fifth grade classroom and facilitated a four-day intervention that consisted of ideas which promote problem solving discourse in a mathematics classroom. The main focus of the study was implementing a modified think-pair-share strategy so that “students feel a sense of belonging in the classroom where mathematical discussions are prevalent (2010)”. The article also provides a broad list of guidelines which support such classroom discourse. These guidelines include a brief description of how Bostic and Jacobbe successfully implemented this intervention in the classroom. The intended audience for this reading is mathematics teachers in any grade level; slight scaffolding and modifications may be intended to better serve particular age groups. This study illuminates effective ways of introducing problem solving as well as a brief introduction to the role of the teacher as a fellow problem solver rather than a mathematical authority. (S. Richards)

Brigham, F. J., Wilson, R., Jones, E., Moisio, M. (1996). Best Practices: Teaching Decimals, Fractions, and Percents to Students with Learning Disabilities.

LD Forum,21(3), 10-15.The authors of this article share best practices for teaching students with learning disabilities (LD) about fractions, decimals, and percents. Unfortunately, there are only a small number of instructional practices that have been investigated for teachers who have students with learning disabilities. Teaching the “Big Idea” will help students with LD better grasp the overarching concepts and themes in the current study. For example, the big idea for a unit covering decimals, fractions, and percents could be studying ratios as a form of division. Once students with LD can begin making connections within the context, the information and materials will make better sense for their learning desires. Included in the article is a general list of teaching techniques specifically advised for teachers of students with LD. The suggestions are unique to this unit of study, but can also be interpreted in more general terms to better suit another unit. (S. Richards)

Burns, M. (November, 2007). Nine Ways to Catch Kids Up.

Educational Leadership: Association for Supervision and Curriculum Development,16-21.

Marilyn Burns, founder of Math Solutions Professional Development, writes about her successes in developing lessons that help intervention students catch up and keep up in learning mathematics. She recalls that these helpful suggestions might not be useful for all students, but for the students who are truly at risk of failure, these things can change the ways in which students can learn and achieve greater success. Three timing guidelines for offering instruction intervention to support floundering students can be before, during, or after the lesson of topic is discussed and taught. Here, Burns mentions pros and cons of each. She also clearly states that providing extra help to struggling learners must be more than just extra practice. Rather, in nine effective and unique ways, teachers can provide extra help to students without doubling the amount of work. Burns includes a short conversation she had with a student who was struggling with multiplication. What the reader is able to learn from this example is that many students are struggling because they lack connections between prior knowledge and have difficulty building a strong foundation of mathematics. (S. Richards)

Campe, K. D. (2011). Do it right: Strategies for implementing technology.

Mathematics Teacher,104(8), 620-625.This article encompasses ways to incorporate technology into a mathematics classroom. Campe offers strategies and tips on how to utechnology “Before The Lesson”, “During The Lesson”, and “After The Lesson” (Campe, 2011). Not only does she give tips for incorporating technology successfully, she also gives real-life examples of how it would work in the mathematics classroom (multiple figures are shown as examples). Many times teachers focus too much on the process of using the technology instead of the teaching benefits that the tese chnology has to offer mathematics students. This article explicitly states what a mathematics teacher needs to do to make technology successful for his or her mathematics students. The big idea Campe has to offer to mathematics teachers is to use the technology that is available to them; and to make good use of it (2011). She also explains how to use technology to help teach mathematics conceptually instead of procedurally. (R. McDowell)

Charlesworth, R., & Leali, S. (2012). Using problem solving to assess young children’s knowledge.

Early Childhood Education Journal, 39(6),373-382.

Problem solving involves students working through four different learning processes: reasoning, communication, connections, and representations. The idea of problem solving is to provide a window into students mathematical thinking and understanding that leads to better ways of assessing. The authors suggest that assessments should be incorporated in everyday activities so that it becomes routine rather than an interruption. The article includes different ways of assessing young children with the goal of answering where a child is at currently. A large portion of this text was for teachers to consider the importance of questioning students regarding their mathematical thinking as they work through a word problem. Three main ways of assessment mentioned in this article are the following: observation, informal conversations, and interview assessment. Throughout the text are examples of student conversations with adults about solving mathematical problems such as probing questions for informal conversations or examples of interview assessments. (S. Richards)

Clausen-May, T., & Lord, T. (2000). Thinking styles and formal assessment: Spatial and

numerical thinkers in the mathematics classroom.Mathematics in School 29(4), 10-13.The authors start off by discussing how two students solved the same problem in different ways. One student solved the problem using an analytical numerical method; the other used a holistic spatial approach. The authors used this problem in addition “to some of the data which had been collected during the development of two series of tests, the

Mental Mathematics 6-14series and theGo Practice True Mental Mathseries,” (p.10) in a study at the National Foundation for Educational Research. During the development of these tests, 1111 twelve and thirteen-year-old students answered seventy trial questions. The authors used these questions to create two, eight question sub-tests, a spatial sub-test and a numerical sub-test. From these sub-tests, they identified students that scored low on the spatial questions, but high on the numerical questions, and low on the numerical questions, but high on the spatial questions. After additional analysis of these two groups, the authors found that overall “spatial thinkers performed less will than numerical thinkers” (p. 13). The authors conclude the article by stating, “Even such ‘spatial’ topics as volume and angle are taught, learned, and assessed in a way that disables the spatial thinkers in our classroom” (p. 13). (L. Gainey)

Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective Discourse and Collective Reflection.

Journal for Research in Mathematics Education, 2(3), 258-277.The authors studied discourse in a first-grade classroom to analyze the relationship between classroom discourse and mathematical development. They focus on reflective discourse, by analyzing two classroom episodes. The children were able to reflect, through participation in discourse, on the previous activity. The author’s propose that reflective discourse creates conditions suitable for mathematics learning, but does not necessarily guarantee it. Although there was collective discourse, it is up to the individual students to reflect and form ideas. In addition to individual students playing an important role in reflective discourse, the teacher is also key. The two major contributions of this teacher were guiding the discussion and representing student’s ideas using symbols. It is important that teachers facilitate the discussion to “initiate shifts” (pg. 269) as needed. In addition, the symbolic representations the teacher used help to guide the discussion. This article provides examples of reflective discourse that make it clear what they are describing. I also agree with the notion that discourse does not cause learning, but allows the possibility of learning. (K. Dwyer)

Cuoco, A, Goldenberg, E. P., & Mark, J.. (1996).

Habits of mind: An organizing principle for mathematics curricula.Journal of Mathematical Behavior, 15,375-407.This is an article about curriculum so it might be dismissed as something not relevant to our discussions of mathematics instruction. That would be a mistake. The central thesis of this article is that the methods by which mathematics is created and the techniques used by mathematics researchers should be the basis for how students do mathematics, think about mathematics problems, and learn mathematics. Some mathematical habits of mind are: STUDENTS should be pattern sniffers, experimenters, describers, tinkerers, inventers, visualizers, conjecturers, and guessers. Many examples and descriptions are given along with discussions of how mathematicians think. Cuoco, Goldenberg, and Mark are mathematicians (in the usual sense of that word) and their presentations are easy to follow. A thought-provoking footnote is "Of course, by mathematicians, we mean more than just members of AMS; we mean people who do mathematics. Some mathematicans are children; some would never call themselves mathematicians." (p. 384) Think about it... (J. Wilson)

Davis, B., & Simmt, E. (2006). Mathematics-for-Teaching: An Ongoing Investigation of the Mathematics That Teachers (Need to) Know.

Educational Studies in Mathematics, 61(3), 293-319.In this article Davis and Simmt. offer a theoretical discussion of teachers’ mathematics-for-teaching, using complexity science as a framework for interpretation. The authors illustrate the discussion with some teachers’ interactions around mathematics that arose in the context of in-service sessions. They use the events from the sessions to illustrate four intertwining aspects of teachers’ mathematics-for-teaching: 1) mathematical objects, 2) curriculum structures, 3) classroom collectivity, and 4) subjective understanding. Authors draw on complexity science and argue that these phenomena are nested in one another and that they obey similar dynamics, albeit on very different scale. Davis and Simmt conclude that 1) a particular fluency with these four aspects is important for mathematics teaching, and 2) these aspects might serve as appropriate emphases for courses in mathematics intended for teachers. (A. Kar)

de Oliveira, L. (2011). In their shoes: Teachers experience the needs of english language learners through a math simulation.

Multicultural Education,19(1), 59-62.Luciana C. de Oliveira writes about the experiences of teachers that completed a mathematics simulations through the eyes of ELL students. The mathematics simulation was completed in Brazilian Portuguese and given to teachers and pre-service teachers in Indiana. The idea is for teachers and pre-service teachers to develop an awareness of what ELL students go through in school. There were two parts to the study: Math Simulation and Teaching ELLs, and In Their Shoes. In the simulation phase, phase one did not use any ESL strategies, the author read and re-read the activity in Brazilian Portuguese. Then she had the participants write out their feelings. Phase two employed ESL strategies, the participants also answered similar questions. In the second part, the author analyzed the participants’ answers and reflections. She found in phase one (no ESL strategies), the participants were frustrated; in the second phase, the participants were more appreciative of the strategies and less frustrated. This shows the need for ESL strategies for ELL students in our classrooms.

(R. McDowell)

Diversity in Mathematics Education Center for Learning and Teaching (DiME). (2007). Culture, race, power, and mathematics education. In F. Lester (Ed.),

Second handbook of research on mathematics teaching and learning(Vol.1, pp. 450-434). Reston, VA: National Council of Teachers of Mathematics.In this article, DiME (2007) capitalizes on the inequities in mathematics education. DiME claims, “By focusing our attention on equity, we deal only with effects while ignoring the causes of the inequity that we see” (p. 423). DiME argues the notion that inequity issues can be represented by particular groups mathematical achievement, which can be paralleled to “sociopolitical organization of mathematics classrooms” (p. 407). For example: some schools possessing majority Blacks and Hispanics do not have access to higher-level mathematics courses to prepare for college. DiME mentions how “The Algebra Project” did not improve marginalized students’ access to higher-level mathematics courses in high school. DiME suggests utilizing equitable teaching approaches such as complex instruction to promote relational equity among students. DiME suggests that culturally relevant pedagogy and teaching mathematics for social justice can enhance marginalized students’ participation and reform their identity as a learner of mathematics. DiME argues how NCTM’s goals for “all learners” is vague and does not effectively mend the issues of inequity in mathematics classrooms. Lastly, DiME discusses the effects of tracking and standardized labels as a form of continued racism. (C. Ramsey)

Dogan, H. (2012). Emotion, Confidence, Perception and Expectations Case of Mathematics.

International Journal of Science & Mathematics Education,10(1), 49-69. http://ehis.ebscohost.com/eds/pdfviewer/pdfviewer?sid=94449a10-5904-400e-ba12-1b46f89e89ab%40sessionmgr115&vid=5&hid=104Dogan compares two groups of pre-service teachers taking a class in rational numbers, geometry, measurement and problem solving. The traditional group learned the material through lecture. The cohort group learned the material through discourse. Each group was given a pre and post-test measuring aspects of their mathematical perception, emotions, confidence, and course expectations. There was some light, non-statistically significant evidence that the cohort’s class encouraged the view that math is not just rote procedures. More pronounced was the improvement in emotion than in perception. Fear, frustration, and difficulty decrased for those in the cohort class, and enjoyment increased. The students in general were confident in their ability to learn mathematics. The cohort group seemed to ‘catch up’ to the traditional group in terms of perception of their current abilities after having taken the class. A notable result was that both groups were significantly less enthused about teaching math after the course. (J. Traxler)

Draper, R. (2002). School mathematics reform, constructivism, and literacy: A case for literacy instruction in the reform-oriented math classroom.

Journal of Adolescent & Adult Literacy, 45(6), 520-529.In the past decade, mathematics reformers are calling for revamped ways of teaching. Draper does a remarkable job of explaining how constructivism can serve as the best avenue in connecting new approaches to teaching and learning through investigation and discovery with literacy instruction in the mathematics classroom. Draper makes a solid argument for utilizing theories grounded in content-area literacy and constructivism to help transform traditional mathematics classrooms. Draper guides our understanding in how mathematics teachers play a vital role in helping develop literate students through the statement: “literacy instruction is inseparable from meaningful math instruction.” Draper gives examples of literacy strategies that enhance students’ ability to read, think, and comprehend that are most effective in mathematics classrooms due to mathematics classrooms being “text-rich environments.” Through this article, it is apparent that through adopting constructivist pedagogy, teachers can create a truly student-centered mathematics classroom that builds on students’ prior knowledge, interests, and skills. (C. Ramsey)

Dunston, P. J., & Tyminski, A. M. (2013) What's the big deal about vocabulary?

Mathematics Teaching in the Middle School, 19(1), 38-45.Professor Dunston, a literacy educator, and Professor Tyminski, a mathematics educator, at Clemson University came together to write this article about how teaching vocabulary instruction to middle school students can enhance conceptual understanding and set the foundation for appropriate mathematical discourse. In this article, they focus on three research-based vocabulary instruction strategies that are very effective in a middle school mathematics class. 1) The Frayer Model is a “graphic organizer [that helps] students identify examples and non-examples of a concept and differentiate between characteristics of a concept; 2) Four Square is “a graphic organizer [that] allows students to associate many ideas with the term being explored;” and 3) Feature Analysis “is used to illustrate relationships between different terms or concepts” ( pp. 41-43). The authors make a strong argument to conclude explicitly teaching mathematical vocabulary is essential for conceptual understanding and appropriate mathematical language for young adolescents. (C. Ramsey)

Dweck, C. (2007). The perils and promises of praise. Early Intervention at Every Age,

Educational Leadership, 65(2), 34-39. http://maryschmidt.pbworks.com/f/Perils of Praise-Dweck.pdfDweck, a professor of psychology at Stanford University, reveals convincingly in this article how teacher praise can hinder student motivation and can actually cause more harm than good. She states “Praise is intricately connected to how students view their intelligence” (p. 1). She goes on to defined the notions of fixed versus malleable intelligence and reveals how many students believe their intelligence is fixed—often because of the way teachers and parents have praised students. From this article, it confirmed by belief in the fact that educators have to instill a growth mindset rather than a fixed mindset about their intelligence. She provides examples of how teacher praise for intelligence can diminish a strong work ethic, reduce confidence, and force students to see mistakes as ultimatums by instilling a fixed mind-set. She also gives examples of how one can go about praising students to be growth-oriented because to help create motivation and resilience (p. 4). (C. Ramsey)

Ediger, M. (2012). Quality teaching in mathematics.

Education,133(2), 235-238.In this article, Ediger stresses the importance of having high quality teachers teaching mathematics. This is due to the fact that students will use mathematics in everyday life. He addresses many things that a mathematics teacher can do to help improve his or her students’ proficiency in mathematics. Teachers should allow for active learning in the classroom, and need to reinforce mathematical vocabulary. Teachers need to understand the mathematical knowledge and skill level of their students. The teacher should also be knowledgeable in the content and know how to improve self efficacy in mathematics. These are just a few of the suggestions that Ediger points out in this article. (C. Foy)

Elk, S. B. (1998). Is calculus really that different from algebra? A more logical way to understand and teach calculus.

International Journal of Mathematical Education in Science and Technology,29(3), 351-358.Elk wrote this article to re-examine the underlying concepts of elementary calculus to show that calculus is not a “new and different form of mathematics” by illustrating the algebraic component of the derivative, the definite integral, and

e(p. 351). Namely, differential calculus is the process of giving meaning to 0/0, the definite integral gives meaning to the indeterminant for infinity multiplied by 0, andeis the idea of raising 1 to an infinite power using limits from the positive side. Elk goes through specific examples of each case, and shows how blindly applying rules (such as never divide by 0) leads to confronting contradictions. Although the article does not go into how to solve the problems, it does shed light on some algebraic origins of the definitions of these elementary calculus concepts. (I. Stevens)

Even, R. (1993). Subject-Matter Knowledge and Pedagogical Content Knowledge: Prospective Secondary Teachers and the Function Concept.

Journal for Research in Mathematics Education, March 1993, 24(2), 94-116.This article investigates teachers' subject-matter knowledge and its interrelations with pedagogical content knowledge in the context of teaching the concept of function. During the first phase of data collection, 152 prospective secondary teachers completed an open-ended questionnaire concerning their knowledge about function. In the second phase, an additional 10 prospective teachers were interviewed after responding to the questionnaire. The analysis shows that many of the subjects did not have a modern conception of function. Appreciation of the arbitrary nature of functions was missing, and very few could explain the importance and origin of the univalence requirement. This limited conception of function influenced the subjects' pedagogical thinking. Therefore, when describing functions for students, many used their limited concept image and tended not to employ modern terms. In addition, many chose to provide students with a rule to be followed without concern for understanding. (A. Kar)

Faulkner, V. N., & Cain, C. R. (2013). Improving the Mathematical Content Knowledge of General and Special Educators: Evaluating a Professional Development Module That Focuses on Number Sense.

Teacher Education and Special Education: The Journal of the Teacher Education Divisionof the Council for Exceptional Children, 36(2), 115-131.There is evidence that strong teacher knowledge and understanding of content impacts student gains on standardized measures. Faulkner and Cain conduct research testing the effects of 40 hours of professional development has on the following:

- Teacher’s understanding of mathematics within the content area and
- Increase the likelihood that teachers will communicate mathematics more effectively and coherently.
The purpose of the training was to provide meaningful professional development to teachers that would translate into stronger classroom practices for those who instruct students with special needs. The 5-day intervention focused on teaching number sense in such a way that improves the quality of interaction between the teacher and students. This study emphasized discussion that helps students develop their own understanding of numbers and gears away from procedural lessons. The research involved many teachers assigned to either the treatment group or either of the two comparison groups. The treatment group consisted of teachers, both general education and special education, who received solely the North Carolina Foundations of Mathematics Training (NCFMT). The comparison group A received NCFMT training and state-training in grade level mathematics, and the comparison group B received no mathematics training during the past seven years. The results supported the positive effect professional development with NCFMT has on the impact of mathematics teachers. (S. Richards)

Fraser, D. (2013). 5 tips for creating independent activities aligned with the common core state standards.

Teaching Exceptional Children,45(6), 6-15.This article, written by Dawn W. Fraser, is geared towards teaching young elementary students. The goal of this article is to give teacher five tips to use when teaching children with exceptionalities (students with disabilities). The five tips listed and explained in the article were aligned to the Common Core State Standards and are intended to be use with children who have moderate to severe disabilities in an elementary school setting. Not only was the teacher teaching these students mathematics, but also life skills. Even though the setting for this article was elementary school, I can see the tips being modified to fit a middle school mathematics classroom (especially tip one). The five tips Fraser offers are the following: 1) Take advantage of mathematics manipulatives; 2) Use unifix cubes in new ways; 3) Get creative with clothespins; 4) Visit your local craft store; and 5) Use grocery store items for academic, daily living and vocational skills. (R. McDowell)

Ganley, C. M., & Vasilyeva, M. (2013). The Role of Anxiety and Working Memory in Gender Differences in Mathematics.

Journal Of Educational Psychology. doi: 10.1037/a0034099This article examined a study that looked at the effect anxiety has on the memory and its relationship it has on mathematical performance. The study specifically looked at gender differences. The research suggests that a woman’s heightened anxiety may utilize their working memory resources, which led to gender differences on the mathematics test. The literature supporting this study examines the affective and cognitive factors that influence gender differences in mathematical performance. It has been shown that gender differences in mathematics are not as visible at first, but the gap widens as students get older. These gender differences tend to favor boys over girls. The research also says that when the mathematics is more complex, the male advantage may be greater. The study goes on to explain anxiety differences in males and females and the different effects these differences have on mathematical performance. (C.Foy)

Gavin, K., Casa, T., Adelson, J., Caroll, S., & Sheffield, L. (2009). The Impact of Advanced Curriculum on the Achievement of Mathematically Promising Elementary Students.

Gifted Child Quarterly,53(188),1-15. http://gcq.sagepub.com/content/53/3/188.full.pdf+htmlGavin, Adelson, Carroll and Sheffield took 3rd 4th and 5th graders that were mathematically gifted according to the Iowa Test of Basic Skills (ITBS) Two experimental groups were put into the Mentoring Mathematical Minds (M3) program and one comparison group was given normal curriculum. The teachers were given a two week workshop to prepare. M3 involved teaching students to think like professional mathematicians by emphasizing rich, concept based learning. The course involves accelerated material in algebra, data analysis, geometry or measurement, and numbers and operations. Students were evaluated using a combination of ITBS and a norm-referenced standardized assessment. The effects of M3 seemed significant, with effect sizes of .29 to .59 for ITBS concepts and estimation scale and .69 to .97 on the open-response assessment. (J. Traxler)

Gay, S. (2008). Helping teachers connect vocabulary to conceptual understanding.

Mathematics Teacher, 102(3), 218-223.Gay, a mathematics professor at the University of Kansas, wrote this article to reveal the importance of using appropriate mathematical language in the classroom. Through this article, Gay describes strategies used in a class she co-teaches with a colleague who specializes in content-area literacy class. Gay describes how utilizing a “concept attainment model of teaching” can allow students to create their own definitions for a concept which builds a deeper conceptual understanding. In addition, Gay describes and provides examples for two vocabulary strategies that she introduces to her pre-service teachers that many take and use in their pre-service teaching experiences: 1) graphic organizers and 2) the concept circle. The graphic organizer she focuses on is the Frayer model, which I described from my article last week. The concept circle Gay describes is meant to “encourage students to study words critically, relating them conceptually to one another” (Gay, 2008, pg. 221). There are many different modifications that can be applied to this instructional tool to benefit conceptual understanding. Gay touches on the usage of analogies in the mathematics classroom and lastly, she discusses how many terms used in mathematics have multiple meanings outside of the mathematics classroom. She puts emphasis on a teacher’s responsibility for explicitly explaining what these terms mean in terms of mathematics. Gay argues that the course offered to pre-service teachers allows these future educators to understand how improper usage of mathematical language can hinder conceptual understanding for students. (C. Ramsey)

Giesen, J., Cavenaugh, B., & McDonnell, M. (2012). Academic Supports, Cognitive Disability, and Mathematics Achievement for Visually Impaired Youth: A Multilevel Modeling Approach.

International Journal of Special Education, 27(1), 17-26.This is a research synthesis of the impact of academic support on the achievement of elementary and middle grade students who are visually impaired (VI). The extent of academic support in schools was positively related in mathematical achievement in students who were visually impaired without any cognitive disabilities. Research suggests that VI students lag behind their sighted peers in mathematics achievement because these students are not receiving adequate support in the regular education classrooms. For these students, academic support may consist of supplemental instruction such as utilizing Braille math, tutoring, and mentoring. To measure the outcome, students with visual impairments were testing which included auditory and visual stimuli and involved students to calculate problems given to them orally. In conclusion of the study, students enrolled with schools that offer more academic support are more successful than those students who are not given adequate support for their disability. (S. Richards)

Gilbert, M. C., & Musu, L. E. (2008). Using TARGETTS to Create Learning Environments that Support Mathematical Understanding and Adaptive Motivation

Teaching Children Mathematics, 15(3), 138-143.Gilbert and Musu suggest using TARGETTS as a lesson planning and analysis tool for fostering a classroom environment that promotes conceptual understanding and adaptive motivation. Students that exhibit adaptive motivation value mathematics and show a more meaningful understanding of mathematics concepts. The acronym TARGETTS is broken down into tasks, autonomy, grouping, evaluation, time allocation, teacher expectations, and social interaction. Gilbert and Musu feel that TARGETTS highlights the important findings in adaptive motivation while also supporting the National Council of Teachers of Mathematics (NCTM) Communication Processes Standards as well as other NCTM standards. The authors also provide a method for implementing TARGETTS in the classroom and a way of analyzing a lesson plan to see if each area is covered. (K. Dwyer)

Glasersfeld, E. von (1980). The concept of equilibration in a constructivist theory of knowledge. In F. Benseler, P. M. Hejl, & W. K. Koeck (eds.)

Autopoiesis, communication, and society. Frankfurt/New York: Campus, 75–85. (Related EvonG article)In this article Ernst von Glasersfeld explains the connection between the concepts of perturbation, equilibration and goal-directedness in the construction of knowledge. He advocates that the Piaget’s theory of cognition is a constructivist theory of cognition in which knowledge is constructed through mental actions and schemes. Schemes consist of three components: a template for recognizing situations in which the scheme applies, mental actions that are triggered when such a situation is recognized and expected results of operating. This article presents a constructivist view of construction of knowledge with various examples from homeostatic devices to sucking action of a new-born. This is an interesting piece for understanding how knowledge is constructed in view of constructivists. (S. Ghosh Hajra)

Glasersfeld, E. von (1981). Einführung in den radikalen konstrucktivismus. In P. Watzlawick (ed.)

Die erfundene Wirklichkeit, ( pp. 16–38). Munich: Piper. English translation: An introduction to radical constructivism. In P. Watzlawick (ed.) (1984)The invented reality, ( pp. 17–40) New York: Norton.This article presents radical constructivism, which according to Glasersfeld is a possible model of knowing and the acquisition of knowledge is through one’s own experiences. The article is divided into three sections. In the first section, Glasersfeld describes the relationship between knowledge and absolute reality. Here he tries to show that our knowledge is not a representation of the real world but it acts as a key that unlocks many paths. In this section, Glasersfeld explains the main difference between radical constructivism and traditional conceptualizations concerning the relation of knowledge and reality. In traditional view of epistemology, this relation is always of correspondence or matching whereas in radical constructivism, the relation is like an adaptation. In the second section, Glasersfeld outlines the beginning of skepticism, the philosopher Kant’s insight and the first true constructivist Giambattista Vico’s thought. In the last section, Glasersfeld describes some of the main traits of the constructivist analysis of concepts. (S. Ghosh Hajra)

Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example.

Educational Studies in Mathematics,39(1), 111-129.

This article considers the Dutch approach of realistic mathematics education (RME) as a method of using context problems as “models for” instead of “models of” mathematical reasoning (111). Instead of a procedural or entirely graphical approach, which the authors believe may leave the students unable to understand the whole picture or unable to connect concepts to formal mathematics, this method suggests that teachers employ guided reinvention, often by considering the historical origins of the concepts. The idea is to keep the gap between where the students are and what is being introduced as small as possible by designing a hypothetical learning trajectory for the students to be able to reinvent formal mathematics. The article gives some examples of how to use this method by considering time, velocity, and distance as the context for learning derivatives. The goals of RME align with some of the ideas that Polya and Freudenthal present. (I. Stevens).

Gutstein, E. (2013). Reflections on Teaching and Learning Mathematics for Social Justice in Urban Schools. In D. Stinson (Ed),

Teaching Mathematics for Social Justice: Conversations with Educators(pp. 63-78). Reston, VA: National Council of Teachers of Mathematics.

In this article, Eric Gutstein summarizes his experience using mathematics for social justice approach in two Chicago-area schools over several years. The author applied his approach in schools where majority of the students were either Latina/o or Black, and from low-income families. Gutstein admits, at the beginning, it was hard to generate any mathematical interest among students. But as soon as he starts to link math to different societal, economical, political etc. scenarios, students started to engage enthusiastically. Gutstein tries to bridge the gap between Mathematics and Social Justice in his classroom. He says, “My point was that life was full of both injustice and mathematics – it was “just” necessary to bring the two together” (2013). Later he had his students research various topics and explore those from mathematical point of view. While doing this, there were some resistance from administrators and challenges of passing the standardized/”gate keeping” tests. But Gutstein was able to generate enough interest in students, parents, and co-workers that it became a very effective tool for those classes’ successes. (A. Kar)

Gutstein, E. (2003) Teaching Mathematics for Social Justice in an Urban, Latino School.

Journal for Research in Mathematics Education, 34(1), 37-73.This 2003 article by Gutstein presents one of his most extensive research projects.

Hansen-Thomas, H. (2009). Reform-oriented mathematics in three 6th grade classes: How teachers draw in ELLs to academic discourse.

Journal of Language, Identity, and Education,8,88-106.Holly Hansen-Thomas from Texas Woman’s University describes ways to get ELL students engaged in the mathematics content and how to get them to participate. Teachers in this article used the Connected Math Project (CMP). Three sixth grade mathematics classrooms were compared. Hansen-Thomas researched the following three questions: 1) “What are the teachers doing

with languageto encourage and increase (discursive) student participation and ultimately academic discourse in the classroom community? 2) What other techniques (apart from language) do the teachers use to draw students into student participation in the classroom community? 3) What is the role of curriculum in the encouragement of student discourse?” (p. 89). One way the teachers got the students involved was through discourse; by modeling and practicing discourse. Hansen-Thomas gave examples of discourse with multiple teachers. She found the following: “The implications that can be drawn from this study suggest that when ELLs are actively drawn in through elicitation of practice, they have more opportunities to engage and participate in the development of successful mathematical discourse” (p. 103). (R. McDowell)

Hassan, N., Ching, K., Hamizah, N. (2012) Gifted Students' Affinity Toward Mathematics.

Advances in Natural and Applied Sciences, 6(8), 1219-1222.

http://ehis.ebscohost.com/eds/pdfviewer/pdfviewer?sid=e56c15ff-e638-4183-9b4f-574b3d46e6c1%40sessionmgr110&vid=1&hid=107This study by Hassan, Ching, and Hamizah investigated the relative positive effect that various factors have on gifted children's attitude toward mathematics. Thirty gifted students from PERMAT Apintar National Gifted Centre students, ranging from low to high in their level of efficacy, were selected for this study. The students are given a survey where they rank the what the most important factors are for them in feeling positive about mathematics. All groups considered teacher attitude and personality very important. The high scorers valued their grades much more than the other groups for making mathematics seem positive and gave no importance to what textbook was used. The low scoring group put much more value on Learning environment, and much less on challenging questions. The article then makes some inferences based on these results, such as the better grades a student has, the more it positively affects their views on mathematics. (J. Traxler)

Henderlong, J., & Lepper, M. (2002). The effects of praise on children's intrinsic motivation: A review and synthesis.

Psychology Bulletin, 128(5), 774-795.The authors define “praise [as a] ‘positive evaluation made by a person of another’s products, performance, or attributions, where the evaluator presumes the validity of the standards on which the evaluation is based” (p. 775). They note and define the two types of motivation—intrinsic and extrinsic. The authors provide literature support on two contrasting stances regarding praise: 1) praise enhances intrinsic motivation 2) praise undermines intrinsic motivation although the authors themselves never take a specific stance on what they themselves believe and support. The authors heavily discuss five conceptual variables influencing the effects of praise on intrinsic motivation they have gathered from researching both perspectives: 1) perceived sincerity, 2) performance attributions, 3) autonomy, 4) competence, and 5) self efficacy. I think one of the most insightful understandings the authors reveal is that “ability (also know as person or trait oriented) praise would encourage performance-goal orientation [while] process (also know as strategy or effort- oriented) praise would encourage a mastery-goal orientation” (p. ???). The authors also slightly note how culture can highly affect praise and motivation. (C. Ramsey).

Hersh, R. (1993). Proving is Convincing and Explaining.

Educational Studies in Mathematics, 24, 389-399.In this article, Hersh explains how proof plays two different roles, one in mathematical research and the other in the classroom. Mathematical proof can convince and can explain. In research, its role is to convince. In a classroom, convincing students is not a problem. Students are always convinced. In a classroom, the primary role of proof is to explain. This article presents three meanings of proof. The first (colloquial) meaning is to test the true state of affairs. The second (mathematical) meaning is an argument that convinces qualified judges. The third meaning is “a sequence of transformations of formal sentences, carried out according to the rules of the predicate calculus” (p. 391). Author argues, “day-to day mathematical meaning of the proof agrees with the colloquial meaning” (p. 392). Here Hersh makes a point that proof in a classroom is a tool for the teacher and class, whereas for mathematician it is a tool of research. (S. Ghosh Hajra)

Herzig, A. H. (2005). Connecting research to teaching: Goals for achieving diversity in mathematics classrooms.

Mathematics Teacher, 99(4),253-259. http://www.nctm.org/publications/article.aspx?id=18961In this article, Herzig discusses why diversity is important in mathematics and what it takes to succeed in mathematics. Herzig’s focus is turned towards the features of the mathematics education itself, rather than the features of the students. When the school “and classroom processes are aligned in a way that facilitates diverse students’ sense of belongingness and engagement with mathematics, we are more likely to achieve the goal of mathematics for all” (p. 256). Herzig cites equitable opportunity, intellectual climate of the classroom, and economic necessity as reasons why diversity is important. When it comes to success in mathematics, Herzig stresses the impact of a “community of practice”. Additionally, “students’ perceptions [of mathematics] and the social implications of succeeding in mathematics” (p. 257) can affect the diversity within mathematics. (L. Gainey)

Horvath, A., Dietiker, L., Larnell, G., Wang, S., & Smith, J. (2009). Middle-grades mathematics standards: issues and implications.

Mathematics Teaching in the Middle School, 14(5), 275-279.This article by Horvath et al. discusses the issues that standards and expectations bring to teachers and students. Many times teachers receives standards and expectations that are hard to read and a struggle to unpack. This article gives examples of how to unpack standards and expectations. The authors refer to expectations as GLE’s (grade level expectations) and are different for each state. One of the issues discussed is the fact that standards and expectations are not continuous through each grade; sometimes many grades are skipped before that standard is mentioned again (p. 277). Another issue discussed is the ambiguity of the standards given to teachers (p. 278). Both of these issues are examined and examples are given on how to incorporate the same standard in each grade and how to unpack the standards and expectations. The examples of standards and expectations are from numerous states such as Texas, Missouri, and Rhode Island. (R. McDowell)

Huinker, D. (1998). Letting Fraction Algorithms Emerge Through Problem Solving. In L. J. Morrow & M. J. Kenney (Eds),

The Teaching and Learning of Algorithms in school mathematics(pp. 170-182). Reston, VA: National Council of Teachers of Mathematics.The author, Huinker, writes about two urban school classrooms that devote their lessons to encouraging fifth grade students to problem solve by inventing their own algorithms. The teachers take on this challenge despite prior concerns about student performance on state-wide testing. Included in that article are five main guiding principles for teachers to consider as they initiate problem solving in such a creative way. Given are multiple scenarios and glimpses of actual student-teacher conversations as this type of problem solving approach was implemented in this fifth grade classrooms. Emphasizing on the big ideas was an important key ingredient for success with this approach. For example, students explored addition and subtraction of fractions by simply devising a way to “rename” fractions instead of just finding common denominators. This study used fraction strips as a tool for students to visual fractions. Students were also encouraged to record their thinking in writing. In this study, these students constructed intuitive qualitative understanding of fraction concepts and operations (p. 181). (S. Richards)

Jebson, S. R. (2012), Impact of Cooperative Learning Approach on Senior Secondary School Students Performance in Mathematics,

Ife PsychologIA,20(2),107-112.

This article is a study on the impact of cooperative learning approach on the students’ performance in mathematics in some selected secondary schools in Adamawa State in Nigeria. The main objectives of this study are: to determine the impact of cooperative learning approach on students’ performance in mathematics and to determine the effect of gender on students’ performance in mathematics using cooperative learning approach. This study employed parallel group design, which consists of two groups: the control group and the experimental group. The cooperative learning approach was used to teach the experimental group while the control group was taught with the conventional lesson method. Using the Mathematics Test of Assimilation, a pre-test and post-test was administered for the both groups. The study finds that cooperative learning approach has significant effect on student’ performance in secondary school mathematics and there is no significant effect between male and female students’ performance in mathematics using cooperative learning approach. (S. Ghosh Hajra)

Johanning, D. I. (2011). Estimation’s Role in Calculations with Fractions,

Mathematics Teaching in the Middle School,17(2),96 – 102.

This article explores the ways that estimation could be used when students learn various fraction operations. This article presents how students use number sense and also algorithm sense as tools to develop their understanding of fraction operations. Here a sixth grader while working on fractions addition and subtraction asked if six-tenths plus six-sevenths would be 12/17. The author suggested that estimation would be a useful tool for exploring the answer for this question. The author argues that the development of fraction number sense is fundamental to the development of fraction operation sense. The author presents ways to encourage and support estimation as part of the development of fraction operations. The author also highlights the role of the teacher where the teacher did not respond with yes or no for an answer but encouraged students to use estimation as one of the tools to understand fraction operations. (S. Ghosh Hajra)

Johnson, D. W., Johnson, R. T., & Stanne, M. B. (2000, May).

Cooperative Learning Methods: A Meta-Analysis.Retrieved November 10, 2013, from http://dc368.4shared.com/doc/xgmbNJpW/preview.htmlThis article analyzes data found in 164 studies conducted about cooperative learning. Cooperative learning was referred to as a generic term for a method of teaching in which students work together to accomplish the same learning goals. Individual students are only able to meet their goals if other members of the group do also. Several specific types of cooperative learning are Learning Together and Alone, Teams-Games-Tournaments, Group Investigation, Constructive Controversy, Jigsaw, Student Teams Achievement Divisions, Complex Instruction, Team Accelerated Instruction, Cooperative Learning Structures, and Cooperative Integrated Reading and Composition. The purpose of this review was to find the amount of research done on particular cooperative learning strategies, the effectiveness of the different methods on increasing academic achievement, and the characteristics of the more effective strategies. The meta-analysis found that studies have been conducted all around the world at all stages of education. All of the different implementation methods produced higher achievement compared to competitive or individualistic classrooms. Thus, any method that a teacher chooses to use, if implemented properly, should produce these results. (K. Dwyer)

Johnson, B., & van der Sandt, S. (2011). “Math makes me sweat” The Impact of Pre-Service Courses on Mathematics Anxiety.

Issues in the Undergraduate Mathematics Preparation of School Teachers,5, Dec 2011. Also known asIUMPST -- The Journal.http://files.eric.ed.gov/fulltext/EJ962631.pdfJohnson and van der Sandt gave pre-service teachers a Revised Mathematics Anxiety Survey (R-MANX) before and after they take a freshman mathematics content course and before and after a methodology course. Of the four groups they tested all had high anxiety scores, but elementary education students had the least anxiety starting out, and were the only ones who had a statistically significant decrease in anxiety from taking the freshman content course aimed at deeper understandings of elementary mathematics. The special education pre-service teachers had the second lowest anxiety to start out with by a slim margin, but seemed to be relatively unaffected the classwork. The other two groups ‘deaf and hard of hearing’ and ‘early childhood’ appeared to mostly be helped by the methodology course. The article points out that there would probably more appropriate courses or interventions for certain groups, namely the special education pre-service teachers. The article implies that teachers with anxiety can pass that anxiety onto their students in a variety of ways, therefore emphasizing the importance of dealing with the pre-service teacher’s anxiety issues appropriately. (J. Traxler)

Kaplan, R., & Alon, S. (2013). Using technology to teach equivalence: Reflect and discuss.

Teaching Children Mathematics, 19(6),382-389. http://www.jstor.org.proxy-remote.galib.uga.edu/stable/10.5951/teacchilmath.19.6.0382.This article focuses on a professional development session that aims to help teachers learn how to utilize NCTM’s illuminations “Pan Balance” as a means for teaching the concept of equivalence. In mathematics, teachers tend to jump straight to numbers and use the = sign as a means of telling the “answer.” Yet through this article, and the professional development session, it is clear that starting with shapes on the Pan Balance site can help students understand the concept of “the = sign represent[ing] a balance between sides verses ‘Here comes the answer’” (p. 384). The article provides two examples of teachers working with a student that helps them assess their student’s understanding of equivalency rather than if they “did the math correctly.” I like how the article takes the audience back to teaching the concept of something that is later supported by procedural fluency. The article reveals that once the teachers evaluated conceptual understanding, they introduced true of false number sentences such as: 7-5=6-4 and had the students write definitive arguments as to why the statement was true or false. This article reveals that “professional development must be done with attention to the mastery of mathematics content embedded in technology, familiarly with the technology tool itself, and understanding of how children’s learning must be best served by particular technology tool in comparison with other means" (p. 387).

Kaput, J., & Roschelle, J. (1997). Deepening the impact of technology beyond assistance with traditional formalisms in order to democratize access to ideas underlying calculus. PME Plenary Paper, University of Massachusetts-Dartmouth. Retrieved From: http://www.kaputcenter.umassd.edu/downloads/products/publications/deepeningimpact.pdf

In this article, Kaput and Roschelle discusses some of the details of his SimCalc Project, which builds and tests software to support learning of the underlying concepts of calculus starting at as young as 6 years old. Students can control simulations and import physical data into the computational environment. Kaput and Roschelle begin with students’ intuitive experience with speed and motion, and as questions are raised, they have found that children spontaneously engage in “interval analysis” by creating sub-events to understand complex mathematical functions. Kaput and Rochelle claim that the technology enables the students to overcome common misconceptions by exploring the software, and the students are able to understand the general idea of integration based on the area under the graph of a function. Kaput and Roschelle understand the importance of the historical contributions made by previous mathematicians, but urges classrooms to keep up with current technology. (I. Stevens)

Karimi A., & Venkatesan S. (2009). Cognitive Behavioral Group Therapy in Mathematics Anxiety.

Journal of the Indian Academy of Applied Psychology,35(2), 299-303. http://medind.nic.in/jak/t09/i2/jakt09i2p299.pdf

Karimi and Venkatesan investigate the affects of cognitive behavioral therapy on mathematics anxiety. The sample was a group of 13-16 year old Iranian students who tested high on mathematics anxiety. The experimental group was administered a fifteen 1.5 hour sessions cognitive behavioral group therapy (CBGT) sessions. They were also given a workbook with a summary of session material with and home practice problems. The program emphasized work on identifying automatic negative thoughts (NAT), coping with NAT, and assertiveness. The post test demonstrated that a significant decrease in anxiety in both the mathematics test and the numerical tasks categories. The paper concludes that this paper and related research indicates that CBGT and other research can be highly effective at reducing mathematics anxiety for both genders. (J. Traxler)

Karp, A., (2010) Teachers of the Mathematically Gifted Tell about Themselves and Their Profession.

Roper Review, 32,272-280.

http://ehis.ebscohost.com/eds/pdfviewer/pdfviewer?vid=3&sid=045a6afa-45cf-4d79-ae21-f6772df97ef9%40sessionmgr114&hid=116

This article interviews twelve accomplished teachers who relay their experiences with teaching in Russian schools for the mathematically talented. This paper describes the academic upbringing of the teachers, either in primarily in pedagogy or mathematics, and how that shaped their experience in the schools. The main thrust of the paper is how the teachers dealt with the challenges of teaching at an elite school, and the process they went through to adequately meet that challenge. The students' efficacy for problem based learning, and its implications for teaching are emphasized in the paper, implying that teachers must go beyond standard problems develop their capacity to improvise. (J. Traxler)

Kitchen, R, Cherrington, A., Gates, J., Hitchings, J., Majka, M., Merk, M., & Trubow, G. (2002). Supporting the reform through performance assessment.

Mathematics Teaching in the Middle School 8(1),24-30.This article discusses an assessment project conducted at Borel Middle School in San Mateo, California. The mathematics teachers at the school worked together to develop performance assessments that “aligned with the curriculum goals and objectives across the entire 6-8 curriculum” (p. 27). The teachers wanted to create assessments that “promoted higher-order thinking” and “require[d] students to apply their knowledge in real-life contexts” (p. 25). Four performance assessments were written for each grade level. The teachers dedicated three afternoons each quarter to grading these performance assessments. The amount of time they spent analyzing these performance assessment allowed them to pick up on student misconceptions. Additionally, the constant collaboration among teachers created consistency, in both instruction and assessment, from classroom to classroom. The teachers found that “student performance consistently improved during the first three years that the [assessments] were administered” (p. 26). The students also seemed to have a positive view of the assessments and often did extra work to receive a score of “exceeds expectations”. (L. Gainey)

Kober, N., & Rentner, D. (2012).

Year two of implementing the common core state standards: states’ progress and challenges. Washington, DC: Center on Education Policy. Retrieved from: http://www.cep-dc.org/displayDocument.cfm?DocumentID=391This is an article on the progress of using the common core state standards and the challenges that schools and teachers have had when implementing them. The authors surveyed each state’s DOE of those that use the common core state standards. The second page of the article lists and explains the findings so far when using these standards. This is the third year that these standards have been implemented. Therefore, the authors wanted to determine the process after the second year. Because of this standards, curriculum maps and materials/resources have been changed and created. The authors state that even though these standards are “rigorous” (p. 2), states “expect to fully implement them by school year 2014-15” (p. 6). This article includes many tables that show the results of surveys from each state and the timeline for implementation of the standards. Kober and Rentner state that a challenge that still exists is aligning tests with the common core state standards. (R. McDowell)

Kohn, A. (2001). Five Reasons to Stop Saying "Good Job!".

Young Children, 56(5),24-28.Kohn, a renowned lecturer and author on human behavior, parenting, and education, discussed in this article the hindrances of praising—as any adult, parent, coach, and/or teacher. In this article, he discusses how there are “many book and articles [that] advice us against relying on punishment [but] you’ll have to look awfully hard to find a discouraging word about what is euphemistically called positive reinforcement” (pg. 1). Through his article, he reveals 5 ways on how “Good Job” can have negative affects on children’s long term behavior: it can 1) manipulate children, 2) create praise junkies, 3) steal a child’s pleasure, 4) cause children to loose interest, and 5) reduce achievement. At the end of the article, Kohn recommends how to “keep in mind our long-term goals for our children [by] watching what we say” by 1) saying nothing, 2) saying what we [as adults] saw, or 3) talk less and ask more (pg. 5). This article is not necessarily geared toward teacher praise, but it brings into perspective how if we do not consider the long term affects of what we say to encourage children, this can negatively affect their actions and behavior. (C. Ramsey)

Krawec, J., Huang, J., Montague, M., Kressler, B., & de Alba, A. (2012). The Effects of Cognitive Strategy Instruction on Knowledge of Math Problem-Solving Processes of Middle School Students with Learning Disabilities.

Learning Disability Quarterly, 36(2), 80-92. http://ehis.ebscohost.com/eds/pdfviewer/pdfviewer?vid=2&sid=6467fb37-e2de-4608-8f3e-8dab7d0f7fc6%40sessionmgr115&hid=105

Krawec, et al. investigate the effects of Solve It! on seventh and eighth grade students with learning disabilities. Solve It! is an intervention program utilizing scripted lessons to teach students a successful mathematics problem-solving process, i.e., read, paraphrase, visualize, hypothesize, estimate, compute, and then check your work. They confirmed previous literature that demonstrated how the program improved answer precision. The new information they researched was how the program increases the number strategies a student learns and their knowledge of effective problem solving strategy. (J. Traxler)

Lang, X. (1999). CAI and the reform of calculus education in China.

International Journal of Mathematical Education in Science & Technology, 30(3), 399-404.

This article presents the development of computer-assisted instruction (CAI) in China in 1999. Although technology has developed significantly since the article was written, it presents the idea of implementing laboratory courses to foster “the scientific exploratory spirit” of the undergraduate calculus students at an engineering school (401). Lang emphasizes the role of the computer as a tool, not a replacement for brainpower, and notes that good problems should encourage the students to think deeply. Lang gives a couple of examples of problems used in the laboratory courses, and encourages collaboration amongst teachers to construct new meaningful problems. Lang wishes to avoid the traditional lecture and memorization of relevant rules that was a common practice in China prior the attempt to reform undergraduate instruction in the early 1990s emphasizing the importance of practical problems. (I. Stevens)

Larson, M. R., & Leinwand, S. (2013). Prepare for More Realistic Test Results.

Mathematics Teacher, 106(9), 656-659. http://www.nctm.org/publications/article.aspx?id=36586This article discuss about implementation of the Common Core State Standards for Mathematics (CCSSM) and possible drop in performance when the new assessment is administered in 2014-15 school year. The reports points out, when compared to international benchmark, as CCSSM require that standards be internationally benchmarked, “the mean eight-grade state mathematics proficiency rate would drop from 62% to 29%, and would drop in each of the 48 states included in the study except for the Massachusetts and South Carolina” (Larson and Leinwand, 2013). Later the article offers some suggestions on preparing for the likelihood of lower proficiency rates for teachers, school administrators, legislators, and students. The authors stress that perseverance is the key for the CCSSM be successful in the long run. Furthermore, we should not jump into conclusion regarding CCSSM based on it’s initial assessment results only. (A. Kar)

Leikin, R., & Zaslavsky, O. (1997). Facilitating Student Interactions in Mathematics in a Cooperative Learning Setting.

Journal for Research in Mathematics Education, 28(3), 331-354.Leikin and Zaslavsky studied the effects of a cooperative small-group learning environment on the amount of student on-task activities in the classroom. They were also interested in the types of interactions that occurred as well as student’s attitudes. The implemented cooperative learning technique involved students working in pairs to study and teach the learning material. Each unit was structured as follows: whole-class introductory lesson, four problem-solving lessons in groups, whole-class summary lesson, and unit test. It was found that this cooperative learning setting increased the amount of explanations offered and questions asked by students to students. The majority of students also expressed positive attitudes about the learning environment. Students felt more comfortable asking their peers for help and were also more willing to offer help. The study concludes that cooperative learning environments promote mathematical communication. (K. Dwyer)

Leikin, R., & Zaslavsky, O. (1999). Cooperative Learning in Mathematics.

Mathematics Teacher, 92(3), 240-246.In this article, a method of cooperative learning known as exchange-of-knowledge is discussed. The exchange-of-knowledge method involves students working in groups to become experts at a task, which involves studying a worked problem within groups and then individually solving a similar problem. Next, experts are paired with other students. This pair works together to explain and work through the task. By the end of the lesson, all students have acted as both an expert and a learner. The article suggests some of the learning outcomes of cooperative learning, such as increased student activeness, increased willingness to ask for help, and more positive attitudes. By using the exchange-of-knowledge method, students communicate mathematically, which improves understanding, encourages active learning, creates a comfortable environment, and helps teachers uncover students’ thoughts. (K. Dwyer)

Leonard, J., Brooks, W., Barnes-Johnson, J., & Berry III, R. Q. (2010). The Nuances and Complexities of Teaching Mathematics for Cultural Relevance and Social Justice.

Journal of Teacher Education,61(3), 261-271.

In this article Leonard et al. argue that culturally relevant instruction coupled with teaching for social justice can motivate marginalized students to learn mathematics. The authors explore the theoretical frameworks underlying culturally relevant pedagogy and social justice pedagogy, present examples of mathematically teachings that reveal the possibilities and challenges associated with these approaches, and offer recommendations to successfully use of these approaches in today’s classroom. Leonard et al. also caution about picking “narrow conceptualizations” and “irrelevant examples” (p. 268), as it does not lead to positive student identity, independent learning, mathematical empowerment, and development of critical thinking. (A. Kar)

Levine, L. E., & Wasmuth, V. (2004). Laptops, technology, and algebra 1: A case study of an experiment.

Mathematics Teacher 97(2), 136-142.Levine and Wasmuth conducted an informal study “to gain insight into how the use of laptops enhances the learning experience of students” (p. 137). The study compared two Algebra 1 classes. Both of the classes were comprised of honor students and taught by Wasmuth. Levine, who was a professor of mathematical sciences, served as a consultant. All of the students in the experimental class were issued laptops for the semester. Wasmuth used Discourse Instructional Delivery and Assessment Software with the experimental class. The discourse software allowed Wasmuth “to follow [her] students’ learning in real time and assess their comprehension while teaching”(p. 138). Similarly, students were able to do and turn in their homework online. This allowed Wasmuth to pin point spots students were having trouble with and come to class prepared. Students also commented on how they found the immediate feedback provided by the software helpful. The authors found that the average final exam and post-test scores of the experimental group exceeded the experimental group by six points. While they acknowledged that this is not a significant difference, they concluded that there are benefits to using laptops in and outside of class. Additionally, Levine commented on how he thought laptops should be used in other mathematics courses as well. (L. Gainey)

Lubienski, S. (2008). Research commentary: On gap gazing in mathematics education.

Journal for Research in Mathematics Education, 39(4), 350-356. http://www.nctm.org/publications/article.aspx?id=17357“The Achievement Gap” is one of the largest target areas of talk in the educational realm. The Research about the achievement gap has led researchers to debate if there is a need to analyze the “obvious” gaps. Lubienski makes a strong argument as to why “gap gazing” is a critical duty of mathematics educators. Her two main arguments are using gap analyses for shaping public opinion and policy and informing mathematic education research and practice. Lubienski claims, “Although research through a variety of perspectives is valuable, it is dangerous for the mathematics education research community to refrain from gap analyses and allow other to speak in our place (pg. 352). In addition, she argues how “detailed analyses of gaps can help researchers and practioners more effectively target their efforts toward equity,[ through] illuminating which groups to target and what aspects of instruction to address” (pg. 353). Toward the end of her article, she suggests future directions for gap gazing through 1) hierarchical linear modeling, 2) cross-classified models, and 3) propensity score matching. The underlying issue to that calls for the need for gap gazing is that until a gap is found and analyzed, improvement, no matter how small or large the gains, will not be noted in terms of underserved and/or marginalized students. (C. Ramsey)

Magdalene, L. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching.

American Educational Research Journal,27(1), 29-63.This article represents a case study of implementing Lakatos’s and Polya’s ideas of “conscious guessing” and “inductive attitude” in a fifth grade classroom by engaging students in a public analysis of the assumptions they made to formulate answers to a problem presented by the teacher (Magdalene). The problems chosen represented “structured problems requiring productive thinking.” These problems typically have multiple routes to a solution since they are not solved by the application of a known algorithm. While the method promoted students to present and to defend their ideas to the classroom, issues with exerting political power over peers and resilience to alter incorrect reasoning occurred. Overall, this was an interesting article to explore how to teach about learning without direct instruction by highlighting conceptual understanding and the process of public mathematical analysis in the classroom. The method can also be applied in higher level mathematics. (I. Stevens)

Manouchehri, A. (2007). Inquiry-discourse mathematics instruction.

Mathematics Teacher, 101(4),290-300. http://www.jstor.org/stable/20876114.The NCTM encourages the use of student inquiry and mathematical discourse. Inquiry based instructions allows for students to come up with mathematical ideas and discoveries on their own. When a teacher uses inquiry based instruction they are creating a mathematical environment where students are given opportunities to expand mathematical investigations. A teacher should be able to connect ideas and facilitate discourse in the classroom. In this article, the author reflects on a project they did in a classroom. The project the students were given was the cereal box problem where they were to come up with displays of cereal boxes with different amounts of boxes. The article goes through the thought processes of each group of students and demonstrates how successful the discourse that took place was in promoting understanding of deeper mathematics. Their goal in the article is to “illustrate how unexpected results and even false solutions offered by students can be used to enrich student learning as well as the existing curriculum” (p.299). (C. Foy)

Martin, D. (2003). Hidden assumptions and unaddressed questions in

Mathematics for Allrhetoric.The Mathematics Educator, 13(2), 7-21. http://libra.msra.cn/Publication/5012940/hidden-assumptions-and-unaddressed-questions-in-mathematics-for-all-rhetoricThis article discusses some of the hidden implications of achieving Mathematics for All in the name of equity. The author is not saying that Mathematics for All is not a worthy goal, but rather that achieving the goal that all students take algebra should not relieve the responsibilities of mathematics educators. Martin discusses issues of the inadequate preparation for taking algebra for underpresented students, the trickle down effect, and the need for a proper social structure to make learning mathematics practical outside of school. Martin encourages critical analysis, reflection, and the use of critical social/race theory, sociology and anthropology of education in addition to the theory and methods of mathematics education. Overall, Martin is concerned that mathematics educators are “more focused on achieving the goal of getting

therethan on the process ofhowto get there” in terms of equity in mathematics education (p. 14). (I. Stevens)

Martin, D. (2012). Learning Mathematics while Black.

Educational Foundations, 26(1-2), 47-66.

McCloskey, A. V., & Norton, A. H. (2009). Using Steffe’s Fraction Schemes.

Mathematics Teaching in the Middle School, 15(1), 44-50.This article is a sequel to the article “Modeling Students’ Mathematics Using Steffe’s Fraction Schemes” (Norton & McCloskey, 2oo8). Here the authors present the next two of the Steffe’s fraction schemes and examples of fifth and sixth grades students’ work with sample tasks for the two fraction schemes. Here one of the mental operations, the splitting operation, which is the composition of partitioning and iteration, is discussed with examples. This article discusses the two fraction schemes: Reversible partitive fractional scheme and Iterative fractional scheme. The main goal of this article is for teachers to use the discussion as a resource for eliciting, interpreting, and supporting their students’ reasoning with fractions (p. 44). This article followed by the previous one will give the reader a quick understanding of some of the Steffe’s fraction schemes. (S. Ghosh Hajra)

McCrory, R., Folden, R., Ferrini-Mundy, J., Reckase, M., & Senk, S. (2012). Knowledge of Algebra for Teaching: A Framework of Knowledge and Practices.

Journal for Research in Mathematics Education,43(5), 584-615.

In this article McCrory et al. define mathematical content knowledge teachers need to have to teach algebra in classrooms and how teacher preparation and professional development efforts could be coordinated to meet the expectations. Furthermore, authors define categories of knowledge and practices of teaching for understanding and assessing teachers’ knowledge for teaching algebra. The three categories of knowledge are i) school, ii) advanced, and iii) teaching. The framework is based on Ball and her colleagues work on elementary school teaching and recommendations of Conference Board of the Mathematical Sciences for secondary school teachers’ knowledge of post-secondary mathematics. On the other hand, the three categories of teaching practices are i) trimming, ii) bridging, and iii) decompressing. The authors emphasize that assessments of teachers’ mathematical knowledge for teaching algebra should include attention to all combinations of the three keys practices and the three domains. (A. Kar)

McKinney, S., & Frazier, W. (2008). Embracing the principles and standards for school mathematics: an inquiry into the pedagogical and instructional practices of mathematics teachers in high-poverty middle schools.

The Clearing House,81(5), 201-210. http://ehis.ebscohost.com/eds/results?sid=a25b94b9-0469-4c25-8fa3-0b9ce626760c@sessionmgr15&vid=1&hid=2&bquery=Embracing the principles and standards for school mathematics: an inquiry into the pedagogical and instructional practices of mathematics teachers in high-poverty middle schools&bdata=JnR5cGU9MCZzaXRlPWVkcy1saXZlThis article is about the principles and standards for mathematics in middle schools that are labeled as high poverty schools. The principles and standards focused on in this article are the ones created by the National Council of Teachers of Mathematics (NCTM). The authors focused on instructional practices in high-poverty middle schools. The authors were concerned that teachers were not using up to date methods to teach mathematics to middle school students. Therefore they completed this study by analyzing the instructional practices of 200 diverse in-service teachers. In the beginning of the study they found that many teachers were very traditional and used the drill and practice method. They found that inquiry-based methods increased student achievement. The author’s results were organized into different sections based on the principles and standards of NCTM. They found that many of the principles and standards (NCTM) were being used, “although some more than others” (p. 208). (R. McDowell)

Middleton, J. A., & Spanias, P. A. (1999). Motivation for Achievement in Mathematics: Findings, Generalizations, and Criticisms of the Research.

Journal for Research in Mathematics Education, 30(1), 65-88.Middleton and Spanias compile the findings of many studies on motivating students in mathematics. They talk about motivation from several different theoretical points of view including: behavioral, attribution and learned helplessness, goal, and personal-construct theories. The authors generalize the findings of studies done in each of these areas and relate them to each other. Though motivation has yet to be linked to a specific cause, findings show strong correlations between motivation and perceived success. Researchers have also found that motivations develop at a young age and are relatively stable, but can be impacted by teachers. By combining ideas from all the theories of motivation, teachers can movtivate students by providing opportunites for students to develop intrinsic motivation through interest and improving students’ mathematics self confidence by allowing every student to feel successful. (K. Dwyer)

Moschkovich, J. (2009) How can research help us understand mathematics learners who use two languages?

Research Brief, National Council of Teachers of Mathematics.http://www.nctm.org/uploadedFiles/Research_News_and_Advocacy/Research/Clips_and_Briefs/Research_brief_12_Using_2.pdfThis article deals with research on bilingual students and English learners in the mathematics classroom and how more research can benefit these types of students. Moschkovich attempts to answer the question of how research on when and why these students use a particular language can enhance mathematical instruction. Some bilingual students prefer to use two languages when performing computations such as adding or subtracting. The studies look at response time and how being bilingual might affect it. It is also helpful to look at research explaining why students switch between languages during mathematical conversation in the classroom. “The type of mathematics problem and students’ previous experience with mathematics instruction may influence which language a student uses (2009).” In conclusion, this article supports the claim that switching between languages during mathematical conversation should be used as a resource for these students to communicate mathematically. (C. Foy)

Niess, M. L. (2005). Preparing Teachers to teach science and mathematics with technology: Developing a technology pedagogical content knowledge.

Teaching and Teacher Education,21(5), 509-523.

In this article, the author explores preservice teachers’ pedagogical content knowledge (PCK) development with respect to integrating technology. According to Niess, how teachers learned their subject matter is not necessarily the way their students will need to be taught in 21st century. She used four components of PCK to describe technology-enhanced pedagogical content knowledge (TCPK). The study examined the TCPK of 22 graduate students in a multidimensional science and mathematics teacher preparation program that integrated teaching and learning with technology. She also presents cases to illustrate the difficulties and successes of student teachers’ teaching with technology in implementing TCPK. According to Niess, if implemented effectively, TCPK can be a very useful tool for teachers to engage their students in more in depth studies in science and mathematics. (A. Kar)

Norton, A. H., & McCloskey, A. V. (2008). Modeling Students’ Mathematics Using Steffe’s Fraction Schemes

. Teaching Children Mathematics, 14(1), 48-54.This article presents some of the Steffe’s fraction schemes and examples of students’ work to indicate the fraction schemes used by the students. This article discusses schemes so that teachers might use them in assessing and understanding their students’ ways and means of operating with fractions. Here the components of a scheme are discussed and it is shown how schemes are different from strategies. Five different mental actions are discussed prior to the introduction of the fraction schemes. The five fraction operations are: unitizing, partitioning, disembedding, iteration, and splitting. Based on these mental actions, various fractional schemes are discussed here. They are: simultaneous partitioning scheme, equi-partitioning scheme, part-whole scheme, partitive unit fractional scheme and partitive fractional scheme. This article will give the reader a quick understanding of the fraction schemes. (S. Ghosh Hajra)

Oberle, E., & Reichl, K. (2013). Relations among peer acceptance, inhibitory control, and math achievement in early adolescence.

Journal of Applied Developmental Psychology, 34, 45-51.For middle grades students (9-14 years), relationships are the heart of positive academic, social, moral, and emotional development. This article aims to combine cognitive, social, and academic learning through a research study that reveals the connections between peer acceptance, inhibitory control, and math achievement in early adolescence. The authors reveal “higher inhibitory control contributes to better social acceptance by peers in the school setting, which in turn has a positive impact on early adolescents’ grades” (Oberle & Reichl, pg. 47). The authors support their claim with research of other authors who said, “Having friends in school significantly predicts higher GPA in adolescence, and that this relationship could be explained by having shared academic experience with in-school peers, identifying with the in-school peer group and therefore feeling more connected to aspect of school life in general than those who have predominately out-of-school peers (pg. 46). The research study was conducted with 56 males and 43 females in 4 different 4th and 5th grade classrooms—27 % of the students were ESL students. From the research study, the “results [indeed] revealed positive and significant correlations among all three variables (pg. 48). In summation, higher inhibitory control leads to more peer acceptance and more math success. (C. Ramsey)

O'Roark, J. L. (2013). The myth of differentiation in mathematics: Providing maximum growth.

Mathematics Teacher,107(1), 9-11.This is a "Sound off" editorial in the current issue of the

Mathematics Teacherjournal. The "Sound off" items reflect the views of the writer and not the official stances of the editorial panel or the NCTM. Jason O'Roark is a provocative middle school teacher from Pennsylvania, teaching in a school with well-respected state test scores. He contrasts approaches to within class differentiation. The usual practice in his school for differentiation within heterogeneous classes for students who reach mastery of the current class topic is to provide extensions such as projects, more word problems, or work sheets with some advanced work on the topic. He challenges whether this best serves the needs of the better students. As an alternative in his class by giving access to higher-level material rather than the extensions of the same level. His argument is that current teaching and assessment practices are based on minimum standards as opposed to maximum growth for every student. He believes current teaching and assessment practices are detrimental since resources and policies are directed to minimums.His "solution," however, is aimed to unlock the potential of all our students through truly customized learning. He is not advocating the practices that seem to work for his classes, but rather a very idealized implementation through computer implemented instruction, using a system that currently does not exist. (J. Wilson)

Pajares, F. (1996). Self-Efficacy Beliefs and Mathematical Problem-Solving of Gifted Students.

Contemporary Educational Psychology,21(4),325-344. http://www.sciencedirect.com/science/article/pii/S0361476X96900259#This paper attempts to understand the relationship between student traits and performance in gifted and regular students. Pajares uses statistical tools and path modeling to determine relationships between cognitive ability (measured by California Achievement Test), sex, GPA, self-efficacy for self-regulation, math anxiety, self-efficacy in math, and math performance for both gifted and non-gifted student groups. Prior to taking a high stakes exam, students were asked to rate their confidence on problems similar to exam problems using a likert scale to measure self-efficacy in math. The performance on this exam was used to measure math performance. A path model is used to demonstrate predicting variables. This model and the author’s research literature are used as evidence for causal relationships. Males were found to be overconfident and females were shown to be somewhat under-confident, but more accurately assessed their abilities. They make other interesting causal theories, such as gifted students having more stable self-efficacy due to it being more based on cognitive ability than past performance. (J. Traxler)

Pettit, S. (2011). Factors influencing middle school mathematics teachers’ beliefs about ELLs in mainstream classrooms.

IUMPST: The Journal,5, 0-6.Stacie Pettit completed a study to determine the factors that influence teachers about ELL students; specifically middle school mathematics teachers. She also wanted to determine the support ELL students need, strategies to teach mathematics to them effectively, and the students experiences in school (Pettit, 2011). The study done was in the form of a questionnaire called: “Mathematics Teachers’ Beliefs about English Language Learner Questionnaire”. This was given to over 400 middle school teachers in Georgia. She found that teachers who received training for ELL students felt that they were better prepared than those who did not. This is a survey given to teachers, so one has to wonder if they were all truthful. However, the nature of the questions suggests that they were. Some of the questions in the study revealed true feelings about ELL students. Ultimately, she found that training was needed to effectively teach ELL students mathematics. (R. McDowell)

Phillips, D., Bardsley, M., Bach, T., & Gibb-Brown, K. (2009). "But I teach math!" the journey of middle school mathematics teachers and literacy coaches learning to integrate literacy strategies into the math instruction.

Education,129(3), 467-472.This article focused on how mathematics and literacy teachers could incorporate literacy strategies into a mathematics classroom. A project was designed for mathematics and literacy teachers. The participating parties were Niagra University and a “high need urban school district” both in the state of New York (p. 468). This project received money through a grant given by the US Department of Education and was specifically tied to middle school teachers. The project consisted of 2 phases: 1) teachers discussed learning needs, goals and concerns; 2) development of strategies and creation of resources needed to increase literacy in mathematics. In the end, the mathematics teachers became more confident when incorporating literacy strategies and saw the need for the strategies in their mathematicss classrooms. They were not able to assess the impact the strategies had as of when the paper was written. Therefore, they could not determine if these strategies improved comprehension and mathematics scores on the state tests. (R. McDowell)

Polya, G. (1945/1957).

How to solve it: A New Aspect of Mathematical Method. Second Edition. Garden City, NY: Doubleday Anchor. [First edition, published by Princeton University Press, 1945; Princeton Paperback Printing of the Second Edition, 1971; Princeton Science Library Printing of the Second Edition, 1988] Available as a PDF download and as a free Kindle book.This is a mathematics reference that should be a part of every mathematics teacher's personal library. First written in German around 1935, it circulated as duplicated but unpublished manuscript and followed Polya as he fled from Europe before the war to his new home at Stanford University. It was published in English in 1945. By that time, Polya was well-known as an excellent mathematics researcher and mathematics teacher.

This is a reference, rather than a textbook. Polya discusses his approaches to mathematics via problem solving. He uses the word "Heuristic" as a field of study, a subfield of logic, meaning to study the method of rules of inventing. Later authors describe Polya's ideas as "heuristics" but it is not a term he uses. For Polya,

Heuristicis a field of study and he descibesHow to Solve Itas a dictionary of the field. After a general overview, the book is orgainised, alphbetically, by 67 short articles.Polya proposes PHASES of problems solving. Other authors call them "steps" but Polya does not use that term. His phases are Understanding the Problem, Making a Plan, Carrying Out the Plan, and Looking Back. His discussions and articles make clear that his ideas of problem solving do not follow the strict linearity usually associated with a set of steps. (J. Wilson)

Reys, R., Reys, B., Lapan, R., Holliday, G., & Wasman, D. (2003). Assess the impact of standards-based middle grades mathematics curriculum materials on student achievement.

Journal for Research in Mathematics Education,34(1), 74-95. http://www.jstor.org.proxy-remote.galib.uga.edu/stable/pdfplus/30034700.pdf?acceptTC=true.This article is about a research study (2 year study) done to determine the impact of standards-based curriculum materials on the success of students. The standards focused on in this research study were created by the National Council of Teachers of Mathematics. It was conducted on eighth grade students in six school districts in Missouri. The researchers were looking at the types of textbooks the students used in school. They compared students who used standards-based textbooks with students who did not use these types of materials. The researchers analyzed the scores of the Missouri Assessment of Performance (MAP) Mathematics Examination to determine if the materials helped students improve. This article also shows tables of the comparisons of scores between schools that used the standards-based curriculum and those that did not. Researchers found that students’ scores when using the standards-based curriculum “equaled or exceeded” the scores of those that did not use them (p. 87). (R. McDowell)

Ross, J. A., Scott, G., & Bruce, C. D. (2012). The Gender Confidence Gap in Fractions Knowledge: Gender Differences in Student Belief-Achievement Relationships.

School Science And Mathematics, 112(5), 278-288.This study discusses the issue of the achievement gap between males and females in mathematids. It reviews some literature suggesting that the achievement gap is disappearing. Other research, however, suggests th3 confidenc gap still continues. The confidence gap between genders in mathematics refers t the male over femaile advantage in self-confidence and willingness to engage in mathematical tasks. This study attemps to explain why the confidence gap is still evident yet the achievement gap is disappearing. Furthermore, this article "draws on social cognition theory and recent researdh on gennder differences in mathematis learning to fram an investigation into the relationship beween gender differences in mathematics achievement and beliefs about self and mathematics learning." (p. 278) They found that even though self-confidence and achievement are definitely linked, we cannot assume that differences in self-confidence will go down when differences in achievement go down. (C. Foy)

Ross, A., & Willson, V. (2012). The effects of representations, constructivist approaches, and engagement on middle school students’ algebraic procedure and conceptual understanding.

School Science and Mathematics 112(2), 117-128In this article, Ross and Willson summarize a study that examined the impact of representations, constructivist teaching approaches, and student engagement on middle school students’ procedural and conceptual understanding. The study involved seven seventh and eighth grade teachers and their students. During the duration of the study, 16 algebra lessons were taught. Data was gathered from pre and post-tests in addition to videos of the teachers’ lessons. These videos were “used to determine the pedagogical tools and strategies used, as well as student engagement in the lesson” (p. 120). The study found some sub-categories of representations, constructivist approaches, and student engagement increased learning, while others decreased learning. For example, the study found that iconic representations, independent thinking, problem-centered lessons, and expression and justification of ideas resulted in a decrease in learning. Overall, “the study provided evidence for the need for constructivist approach to teaching [for students to have a] thorough knowledge of procedural and conceptual understanding” (p. 127). (L. Gainey)

Scher, D. (1997). Dynamic visualization and proof: A new approach to a classic problem.

Mathematics Teacher 96(6), 394-398In this article, Daniel Scher discusses the Interactive Geometry (IG) labs that were tested by Best Practices in Education. These labs were developed by both U.S. and Russian teachers and were designed to “introduce teachers in the United States to effective mathematics teaching practices from abroad” (p. 394). The IG labs combined visualization and hands-on exploration techniques from Russia, with dynamic software investigation and deductive reasoning techniques from the U.S. As a result; all of the labs follow the same structure of progression. This progression is evident in the Pirate Problem lab presented in this article. The adapted problem was designed so that students would have to use both algebra and geometry to solve the problem. After implementing the Pirate Problem in a classroom, the teachers and author concluded that there are “benefits of combining exploration with deductive reasoning” (p. 398). They also stress the importance of “bridg[ing] the gap between visual evidence and formal proof” (p. 394). (L. Gainey)

Schiefele, U., & Csikszentmihalyi, M. (1995). Motivation and Ability as Factors in Mathematics Experience and Acheievement.

Journal for Research in Mathematics Education, 26(2), 163-181.Schiefele & Csikszentmihalyi are interested in answering three questions: Does the quality of the experience of doing mathematics depend more on ability or motivation? Does interest in a particular subject better predict outcomes of experience and achievement than general motivation? Does motivation and quality of experience predict achievement independent of ability? Researchers studied 108 freshmen and sophomore students with high ability in at least one subject area. The study found that the quality of a student’s experience in a mathematics class depended more on interest than achievement motivation and not at all on ability. Higher achievement was correlated to higher mathematic ability, while interest did not play much of a role at all. Results also show that interest and experience are related, as well as interest and achievement. The relationship between interest and experience, however, is stronger. There are many suggestions made for future research based on this study. The results do provide us with some valuable information and suggest that increasing students’ interest in mathematics may lead to increased quality of experience, achievement, and motivation. (K. Dwyer)

Schmidt, W. H., & Burroughs, N. A. (Spring, 2013). Springing to life: how greater educational equality could grow from the common core mathematics standards.

American Educator, 2-9.This article by Schmidt and Burroughs highlights the possibility of the Common Core State Standards in Mathematics making education more equitable for all students. They explain in detail the 5 threats of implementing the Common Core State Standards for Mathematics: 1) Local control of the curriculum; 2) Teachers; 3) Textbooks; 4) Assessments; and 5) Parents and Voters. The beginning of the article discusses how the standards implemented before the Common Core State Standards forced tracking in schools and how this was not beneficial to the students. They also mentioned that, in a way, it set them up for failure. They then moved to how the Common Core State Standards could make learning for students more equal. The authors see these standards as having “three characteristics that distinguished the standards of high-achieving nations: focus, rigor, and coherence” (p. 4). The authors also mentioned that these new standards forced new ways of teaching the material and assessing the material. Lastly, Schmidt and Burroughs discuss strategies to successfully implement the new standards. (R. McDowell)

Schoenfeld, A. H. (1988, Spring).

When good teaching leads to bad results: the disasters of "well taught" mathematics classes.Educational psychologist,23(2), 145-166.This article reports on observational research where SAH gathered data from a high school geometry course for a full year. The teacher was experienced and by all indications the course was well taught. SAH looks more critically at what the students might have learned and presents a summary that from a certain mathematical perspective, the course "may have done more harm than good." (p. 145) This an interesting and well done piece of research yet it underscores how the perspective of the researcher becomes very much a part of the process. (J. Wilson)

Schoenfeld A. (1989). Exploration of Students’ Mathematical Beliefs and Behavior.

Journal for Research in Mathematics Education,20(4), 338-355. http://www.jstor.org/stable/749440

Schoenfeld takes a sample of 230 high achieving 10 to 12th grade mathematics students. His team observes their classes and gives them a survey to determine their attitudes toward success in mathematics, classroom mathematics, and their attitudes toward mathematics. At the forefront of the articles discussion was the seeming disconnect from classroom practices of the students and their beliefs about the subject itself. The students considered most memorization of rules and doing two minute problems to be at the forefront of learning mathematics. However, they also believed that mathematics was great for strengthening their creativity and logical skills, which Schoenfeld identifies as a stark contradiction. He posits that this disconnect between classroom and subject perceptions is due to rhetoric the teachers espouse about the merits of doing mathematics. He claims that more admirable goals have been set and talked about in the classroom, but little has been done to actually make the learning of mathematics a deeper experience for students. (J. Traxler)

Silver, H. F., Brunsting, J. R., & Walsh, T. (2008). A User's Guide to Learning Styles and Math Tools. In

(pp. 1-15)Math tools, grades 3–12: 64 ways to differentiate instruction and increase student engagement,.Thousand Oaks, CA: Corwin.As the authors of this book were coming to an end of their careers as mathematics instructors and administrators, they wanted to further explore what a tools based approach to mathematics instruction would look like. So, all three authors agreed to write a book that offered mathematics specific tools to use along with the four distinct styles of instruction. The four styles of instruction included in this text are the following: Mastery style, Understanding style, Self-Expressive style, and Interpersonal style. This book is to be a repertoire of instructional tools that teachers could refer to in order to better differentiate for students various learning styles. Included in chapter 1, the authors previewed four different classroom teachers teaching the same topic. All classrooms had similar goals, but each classroom presented the content in different ways that aligned with each of the four learning styles. Mentioned throughout the text, it is important to help teachers diversify their practices; therefore, this book provides suggestions and ideas to assist teachers as they instruct students with various ways of learning. (S. Richards)

Simon, M. A. (1995) Reconstructing mathematics pedagogy from a constructivist perspective.

Journal for Research in Mathematics Education, 26(2), 114-145.In this article, Simon discusses a constructivist model for teaching. He justifies the need for a model by discussing how the ideas of constructivism alone are not enough to implement constructivist teaching. This model was based off of his pre-existing notations of social constructivism and the data he gathered from a three-year teaching experiment. In addition to examining “students’ developing conceptions [in the experiment, he also focused on] the decision making of the teacher/researcher in [relation] to posing problems” (p. 122). The model he developed,

The Mathematics Teaching Cycle, illustrates the relationships between a teacher’s knowledge, the hypothetical learning trajectory, and assessment of students’ knowledge. The hypothetical learning trajectory outlines a path from a teachers learning goals, to their planned learning activities, to their hypothesis of the learning process. A key to this model is modification. As students’ conceptual understanding is changing, the teacher’s conceptual goals for their students should change as well. (L. Gainey)

Smith, J. (1999). International perspectives: Active learning of mathematics.

Mathematics Teaching in the Middle School 5(2), 108-110.In this article, Smith performs an analysis of mathematical learning activities using a constructivist framework. Quoting

Better Mathematics,“ Mathematics can be effectively learned only by involving pupils in experimenting, questioning, reflecting, discovering, inventing, and discussing.” Smith discusses how it is the teacher’s responsibility to provide activities that create these opportunities. However, it is important for teachers to select “activities that focus the attention of the learner on constructing the intended learning outcomes of the session.” He also discusses how presentation, pupil activity, reflection, and socialization are key when it comes to implementing successful learning activities. The presentation of the activity should grab the students’ attention by presenting them with a challenge, a surprise, and cognitive conflict. Pupil activity should require students to use higher order thinking skills. Students should also be able to reflect on what they have learned to be able to “integrate it with their other mental systems”. Lastly, Smith discusses the importance of socialization and how it allows students to effectively communicate about the mathematics they have learned. (L. Gainey)

Soucie, T., Radovic, N., & Svedrec, R. (2010). Making Technology Work.

Mathematics Teaching in the Middle School,15(8), 466-471.The authors in this article mention several benefits of using technology in the classroom. For example, technology can motivate students and can help them visualize the mathematics. Technology often involves real-life application of mathematics. Before using technology in the classroom, the article identifies some purposes for using it appropriately. This article also provides examples of how certain technology was used in the classroom; student remarks were included to provide the reader with a bright picture of the successes technology can bring into the classroom. One of the examples included in the text was an activity where students worked in pairs – one individual would calculate by hand while the other individual could use a calculator. What really surprised students were actually testing that mental math can sometimes be much easier and faster than relying on a calculator. Following this activity, teachers now have a stable opportunity to discuss the best opportunities to use technology and when not too. Throughout this article, it is apparent that technology should improve the quality of mathematics taught. (S. Richards)

Steffe, L. P. (2010). A New Hypothesis Concerning Children’s Fractional Knowledge. In L. P. Steffe & J. Olive,

Children’s Fractional Knowledge(pp. 1-13). New York, NY: Springer.Here Steffe talks about three hypothesis concerning children’s fractional knowledge. The first and the second hypothesis existed in the literature. Steffe came up with the third hypothesis. The first one is separation hypothesis where study of whole number is considered in the context of discrete quantity and the study of fraction is in the context of continuous quantity. The second hypothesis is the interference hypothesis where it is believed that whole number knowledge interferes with the learning of fractions. The third (

New)hypothesis is that “children’s fractional knowledge can emerge as a reorganization of whole number knowing” (p. 2). Steffe call this hypothesis as reorganization hypothesis because “if a new scheme is constructed by using another scheme in a novel way, the new scheme can be regarded as a reorganization of the prior scheme” (p. 1). He then describes two basic ways to understand the reorganization of the prior scheme. (S. Ghosh Hajra)

Steffe, L. P. (2010).

Perspectives on Children’s Fraction Knowledge. In L. P. Steffe & J. Olive,Children’s Fractional Knowledge(pp. 13-25).New York, NY: Springer.

This article deals with three parts. The first part presents the constructivist perspective of school mathematics. Here he talks about the difference between invention and construction. According to Steffe, invention is the production of unknown by the use of imagination without social interaction, whereas construction involves interaction. In the second part, Steffe describes first- and second-order mathematical knowledge. First-order knowledge refers to our own knowledge and the second order knowledge is the model that the observer constructs of the observed person’s knowledge. Steffe then relates these with children’s mathematics and mathematics of children. Children’s mathematics constitutes children’s mathematical realities. It is indicated by what they do as they engage themselves in mathematical activities. Mathematics of children refers to models of children’s mathematics that an observer develops. The third part deals with the structure of the scheme. Steffe’s structure of scheme is bidirectional whereas von Glasersfeld’s is unidirectional. (S. Ghosh Hajra)

Stein, C. C. (2007). Let's Talk: Promoting Mathematical Discourse in the Classroom.

Mathematics Teacher, 101(4), 285-289.Catherine Stein essentially provides a step-by-step guide to facilitating mathematical discourse in your classroom, as well as a way of measuring the quality of the discourse. The first step involves encouraging students to participate by showing them that you are not just looking for a correct answer, but also for conceptual understanding. After students are aware of your expectations, you can begin working on classroom discourse. Discussions must not only promote conceptual understanding, but must also show that we can learn from mistakes. In order to keep students engaged, teachers must be supportive and encourage collaboration and persistence despite incorrect answers. Lastly, Stein provides a rubric with levels of discourse to help teachers assess discourse. This article was a great introduction on promoting classroom discourse. (Dwyer K.)

Strain, P. S., & Joseph, G. E. (2004). A not so good job with "good job": A response to kohn 2001.

Journal of Positive Behavior Interventions,6(1), 55-59. http://nhcebis.seresc.net/document/filename/354/The_Use_of_Praise_-_A_Response_to_Kohn.pdfStrain & Joseph, professor and assistant professors in educational psychology at the University of Colorado react to Kohn’s article

Five Reasons to Stop Saying "Good Job!"in a very negative way. They evaluate each of his 5 reasons to stop saying good job and critique where his philosophy is flawed and potentially detrimental to children’s growth and development. They suggest much literature supports positive reinforcement strategies and offer an alternative perspective on each of his 5 reasons on how a positive reinforcement strategy would be more beneficial than what Kohn suggests. However, the authors do state “persistent, continuous positive reinforcement is not and should not be the norm” (pg. 56). Additionally towards the end of the article, the professors suggest he is making an underpaid, undervalue job much harder and boldly state, “Kohn’s piece is particularly detrimental to individuals who work to close the achievement gap for children from impoverished school settings and for children with special needs” (pg. 58). (C. Ramsey)

Suurtamm, C. (2012).

Assessment can support reasoning and sense making.

Mathematics Teacher 106(1),28-33. http://www.nctm.org/publications/article.aspx?id=33620In this article, Suurtamm describes how the use of formative assessments is beneficial to both teachers and students. “Formative assessment provides teachers with a window into students’ mathematical reasoning and sense making and creates a forum for student and teaching discussion” (p. 28). Students are able to internalize the feedback they get from these assessments and make adjustments to their existing conceptions. Suurtamm draws on the Curriculum Implementation in Intermediate Math research project to describe what formative assessment looks like in a mathematics classroom. The teachers that participated in this study “were chosen because [their classrooms] promoted mathematical inquiry, investigation, and problem solving” (p. 29). For this article, Suurtamm discusses what occurred and what types of formative assessments were used in three different classrooms. The teacher in the first classroom utilized student monitoring and had students present their findings. The teacher in the second classroom used clickers and promoted student discussions. Lastly, the teacher in the third classroom posed thought provoking questions that allowed students to extend their scope of understanding. (L. Gainey)

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity.

Educational Studies in Mathematics,12, 151-69.

This article discusses how a disconnect or conflict between concept images and concept definitions may result in cognitive conflict factors. Tall and Vinner use limits of sequences and continuity as examples. Concept images of limits and continuity are typically introduced and built up without formal definitions (using the School Mathematics Project Advanced Level texts). Students often create a concept image of a limiting process by which regular formulae never actually equal the limit, which causes conflict in a couple of examples that the authors cite as results of a questionnaire for mathematics students arriving at University. The authors also mention how a weak understanding of the concept definition makes formal proofs difficult for students. Similar issues arise when students concept image of a continuous graph as being a function “drawn” without gaps. In particular, analysis course topics make it difficult for students to visualize the concepts as mental pictures, and a weak understanding of the concept definition may find these situations difficult. (Stevens)

Tall, D. (1986) A graphical approach to integration and the fundamental theorem

.Mathematics Teaching,113,48-51. http://wrap.warwick.ac.uk/498/

This article discusses how the author, Tall, moved from written calculation, to graphing calculator, to the software AREA to discover the pattern of integration. After the students learn the techniques of building up areas with small intervals by hand, and larger intervals via the graphing calculator, Tall directs the attention of his students to the computer. Using the graphs from partial sums, the students are guided to guess possible general area functions and interpolate known points. Tall gives multiple examples to use this type of method to determine positive and negative areas, the fundamental theorem, and the role of continuity. Tall ends the article by promoting the graphical approach to calculus by saying that it is “not just a ‘simple way’ for beginning students, it also provides insight into powerful theorems that occur much later in formal mathematical analysis” (p. 13 of PDF). (I. Stevens)

Tatar, E., & Dikici, R. (2009). The effect of the 4MAT method (learning styles and brain hemispheres) of instruction on achievement in mathematics.

International Journal of Mathematical Education inScience and Technology, 40(8),1027–1036.The authors of this article state that efficient learning can be achieved by matching students’ learning styles. This article discusses a study about the effectiveness of the 4MAT system – eight step cycle of instruction that capitalizes on individual learning styles and brain dominance. The left side of the brain is systematic – analysis and planning are key strategies. The right side of the brain is visual and holistic – patterns and connections are key strategies. Those who reviewed the study reported that when students were instructed through their preferred learning style, they demonstrate: : (i) statistically significant improvement in their attitudes towards instruction, (ii) increased tolerance for cognitive diversity, (iii) statistically significant increased academic achievement, (iv) better discipline/behavior, (v) greater self-discipline for homework completion. In conclusion to this research, the early lessons in each school year must be prepared by taking into account of different learning styles (determined from learning style inventories). (S. Richards)

Tatsis, K., & Koleza, E. (208) Social and socio-mathematical norms in collaborative problem-solving.

European Journal of Teacher Education, 31(1), 89-100.

Tatsis and Koleza paired up 40 university students in Greece and give them mathematical problems to work on. During three one hour sessions they were instructed to work on these problems independently while communicating as much of their thought process to their partner as possible. The study's intent was to determine the social norms that would be created out of this type of interaction by using a symbolic interactionism framework and some conversation diagramming and analysis. The study determined that adult students were quite capable of generating the norms needed to for this sort of collaborative learning activity and observed many interesting norms with the caveat that this study took place in Greece and is therefore subject to its cultural influence. (J. Traxler

Tomlinson, C. A. (1999). Mapping a Route Toward Differentiated Instruction.

Educational Leadership, 57(1), 12-16.This article compares two different classrooms. Both teachers are covering the subject of ancient Rome. One classroom is teacher centered and is mostly lecture based. His students are disengaged, but memorize facts and do well on quizzes. The second teacher believes she is running a differentiated classroom. Her students are having fun and learning about a different topic on ancient Rome. Although her students are enjoying class, the second teacher has no end goal. Tomlinson’s classroom illustrations paint two very different pictures. Neither teacher has an effective learning environment. She then introduces a third teacher. This final classroom illustrated gives vivid descriptions of a teacher who effectively uses differentiation by having an “end goal”, which Tomlinson feels is the key. This article gives a good idea of what differentiation should and should not look like. (K. Dwyer)

Truxaw, M. P,. & DeFranco, T. C. (2007). Lessons from Mr. Larson: an inductive model of teaching for orchestrating discourse.

Mathematics Teacher, 101(4), 268-272. http://www.jstor.org/stable/20876110Mathematical discourse is an important part of instruction as contunually indicated by the NCTM. There is often discrepancy of what type of discourse is most useful. This article introduces the two types of discourse: univocal and dialogic. The article proposes “a strategic mix of univocal and dialogic discourse that, when used in conjunction with an inductive model of teaching, can promote mathematical understanding in students” (2007). The article follows a teacher in an eighth grade algebra class and includes the discourse that took place in one of his lessons. People most often think that promoting mathematical discourse means doing away with “telling”, but this article shows that both univocal and dialogic discourse are useful in generating mathematical understanding. (C. Foy)

The "inductive model" was the label given by Truxaw and DeFranco in their observations of this teacher. At several points, they discuss the teacher's actions as developing "metacognition." See Flavell (1979) and Kuzle (2011) in our references list for discussion of metacognitions -- roughly descrobed as thinking about thinking, or the thinking we do to plan and guide our cognitive activities. Another point made by Truxaw and DeFranco is that the teacher's disposition toward mathematics -- activing

doingmathematics -- may influence the classroom discourse. This is an interesting point. What research is there to demonstrate any relationship between the teacher's disposition toward mathematics and creating a discourse oriented classroom? (J. Wilson)

Vialle, W., Ashton, T., Carlon, G., & Rankin, F. (2001). Acceleration: A Coat of Many Colours.

Roeper Review, 24(1),14-19.This article provides a summary of three types of acceleration, teacher and administrative attitude towards it, and the effects of acceleration. The first research article summarized is over early entry, where students enter kindergarten early, the second involves students skipping grades and the third covers vertical timetabling, where a student is accelerated in a specific subject. The administrators expressed concerns over the first two types of acceleration, partly over concerns of maturity level and sociability. The studies provide evidence that these acceleration programs result in improved social/emotional health, and academic satisfaction for students that would be otherwise under-challenged or meet the criteria for accelration programs. (J. Traxler)

Viholainen, A. (2008). Incoherence of a concept image and erroneous conclusions in the case of differentiability.

The Montana Mathematics Enthusiast,5(2&3), 231-248. http://www.math.umt.edu/tmme/vol5no2and3/TMME_vol5nos2and3_a6_pp.231_248.pdf

In this article, Viholainen considers the results of an interview done with a university student who made conclusions contradictory to the formal theory of mathematics. Viholainen uses the students’ reasoning to consider the relationship between deficiencies in the coherence of the concept image and erroneous conclusions in the case of differentiability. After discussing the definitions and relationships of the ideas of concept definition, concept definition image, formal concept definition, and personal interpretation of the formal concept definition based on the work of Tall, Vinner, and Pinto, and reviewing previous studies, Viholainen discusses the methodology and results of his interview. The student was given four piecewise functions and their graphs and was asked to determine whether the functions were continuous and/or differentiable. Apparent initial incoherence of the concept image before even given the functions led to confusions and contradictions as the student attempted to justify often erroneous methods he used to answer the questions. Analysis revealed that erroneous conclusions were consequences of erroneous ways to connect the pieces of knowledge and that misconceptions and erroneous conclusions may lead to cognitive structures, which may be internally coherent, but whose basis is erroneous. (I. Stevens)Wachira, P., Pourdavood, R. G., & Skitski, R. (January, 2013). Mathematics Teacher’s Role in Promoting Classroom Discourse.

International Journal for Mathematics Teaching and Learning. (IJMTL publishes only in electronic format)

The traditional teaching methods for mathematics tend to consist of the teacher telling the students how to do the math. Recently, the NCTM has made great strides to move away from these practices. They say that a teacher’s main focus should be trying to get their students to communicate mathematically. In this study a high school mathematics classroom was observed and the teacher implemented four strategies thought to improve his students’ mathematical discourse. These strategies included: establishing expectations, using mathematical language, establishing a mathematics community, and establishing formal discourse within the classroom. The study also attempted to examine student reactions to this change in instruction since most students are used to the traditional lecture based instruction. (C. Foy).

Walshaw, M., & Anthony, G. (2008). The teacher’s role in classroom discourse: a review of recent research into mathematics classrooms.

Review of Educational Research, 516-551. http://www.jstor.org/stable/40071136, 78(3)

I found this article to be the most useful article that I have looked at thus far. It supports the use of classroom discourse in the mathematics classroom because studies show that classrooms where students are given opportunities to explain, defend, and argue mathematical ideas create more productive learning environments. The authors provide a critical analysis of the “pedagogies that, through classroom discourse, contribute to students’ active engagement with mathematics” (2008). They emphasize on the quality of classroom discourse in classrooms and not just the quantity. The article also includes implications for mathematics teachers such as what discourse works, how it works, and why it works. (C. Foy).

Walmsley, A. L. E., & Muniz, J. (2003). Cooperative Learning and Its Effects in a High School Geometry Classroom.

Mathematics Teacher, 96(2),112-116.Cooperative learning involves four main aspects: positive interdependence, individual accountability, interpersonal skills, face-to-face promotive interaction, and processing. Positive interdependence means that each student is dependent on other members of the group. Group goals encourage students to work together, learn from one another, and to help each other learn. By also holding individuals accountable ensures that all students are learning. When students care about the success of the group, they are more likely to listen to the ideas of other students. Students become better listeners and are more open to alternative solution methods. While working in a cooperative learning setting, students interact with their peers and explain their reasoning to each other. This not only helps to develop social skills in communicating mathematics, but it also improves conceptual understanding. When students are explaining the methods they used to solve a problem, they are forced to look at the reasoning they used. Muniz and Walmsley state that research has shown “increased academic achievement, better communication skills, and successful social and academic group interactions.” (p. 113) (K. Dwyer)

Wang, L., Beckett, G., & Brown, L. (2006). Controversies of standardized assessment in school accountability reform: A critical synthesis of mutlidisciplinary research evidence.

Applied Measurement in Education, 19(4), 305-328.

This article provides pro, con, and synthesized arguments for standardized assessments on four topics: assessment-driven reform, standards-based assessment, assessment-centered accountability, and high-stakes consequences. The authors use international rankings, SAT, ACT, NAEP data, and psychological theories to provide arguments for each of the views. At the end of the document, the authors provide and action research agenda in efforts to improve the current system of standardized assessment. Although not thoroughly discussed, several sections include a social justice undertone. For example, phrases such as “because students vary markedly in their capabilities, presenting them all with the same tasks under the same conditions would…” (313), “NCLB’s stipulation that all children must reach the same set of high standards at the same time fails to acknowledge the diversity and pluralism embodied in our genes and embraced in our society”(314). (A. Kar & I. Stevens)

Watt, H. M. G. (2005). Attitudes to the use of alternative assessment methods in mathematics: a study with secondary mathematics teachers in Sydney, Australia.

Educational Studies in Mathematics, 58,21-44.This article is a study that focuses on teacher’s attitudes and beliefs about alternative assessment methods in mathematics. Traditional mathematical tests are usually not well-written and mostly test routine skills and restricted computations. These tests consist of repetition of learned procedures. Therefore, there is a need to explore alternative methods of assessment. There are differences in learner characteristics which implies that over-reliance on one form of assessment can alter student success by hindering students’ progress who are able to display their knowledge in other ways. The research conducted in this study suggests that teachers are adequately satisfied with traditional testing as revealed through a survey. Some of the most common alternative assessments that teachers in this study use are observations, oral, and practical tasks used by teachers and shown throughout students’ classwork. Included in the results of this study were common themes that teachers mention as a factor in not using alternative ways of assessment which include the following: time constraints, unstructured nature, insufficient resources, and unsuitable. (S. Richards)

Webb, N. (1993). Collaborative Group Verses Individual Assessment in Mathematics: Group Processes and Outcomes.

CSE Technical Report 352.Los Angeles, CA: National Center for Research on Evaluation, Standards, and Student Testing.

This report is on the study of comparing performance in small-group and individual assessment to determine how well achievement scores from group work represented the skills of individual students. This study also focuses on to determine what additional information about students’ skills was provided by the group dynamics and group-solving processes. The sample for the study was the students of two seventh-grade general mathematics classes in an urban middle school. Students worked collaboratively for one class on decimal numbers operation and two weeks later, students worked on a similar problem individually. The results of this study shows that the performance in the group setting was more than in the individual setting. Study showed many students used the resources of the group to get the right answer without understanding. The study also showed that data on group processes gave important insights into students’ mathematics skills and their behavior in collaborative groups. (S. Ghosh Hajra)

Weissglass, J. (2001). Inequity in Mathematics Education: Questions for Educators.

The Mathematics Educator, 12(2), 34-39.In this article, the author stresses that student’s mathematics understanding/learning is dependent on multiple factors, in contrast to the popular belief that the learning is only concentrated between student and teacher. Weissglass analyzes the mathematical learning issue from the perspective of race and power. He illustrates several examples of race and power affecting learning in a negative way and later explores whether raciest/classiest issues could be eliminated from schools. To support his concern, Weissglass cites several examples from textbook, classroom, and society around students that shows racism and classism is still existence in our education system and it needs to be fixed. The author poses a series of questions throughout the article and analyzes them from different perspectives. This article will help reader to think critically about the issues that are sadly still in existence, but sometimes is very subtle manner. (A. Kar)

Westbrook, A. F. (2011). The Effects of Differentiating Instruction by Learning Styles on Problem Solving in Cooperative Groups. A Thesis submitted in partial fulfillment of the requirement for the degree of Master of Education in Curriculum and Instruction, LaGrange, Georgia.

This study is done to improve students’ problem solving skills and attitudes when working on lengthy performance tasks in cooperative groups through differentiated instruction. In this study, two groups, a control group and a treatment group, from ninth grade, who ranged between fourteen to sixteen years of age, have been studied for 15 days. The students in the control group were grouped in random cooperative groups and the students in the treatment group were grouped by learning styles (auditory, kinesthetic, and visual). “The findings contribute to knowledge and to practice by providing evidence that differentiating by learning style in which students are afforded opportunities to work with one learning style does not improve learning of problem solving standards or attitudes towards problem solving” (p. 72). The result of the study showed that the treatment group did not show significant gains compared to the control group in random cooperative groups. (S. Ghosh Hajra)

Whicker, K. M., Bol, L., & Nunnery, J. A. (1997). Cooperative Learning in the Secondary Mathematics Classroom.

The Journal of Educational Research, 91(1), 42-48. Retrieved 12/5/2013, from http://web.ebscohost.com.proxy-remote.galib.uga.edu/ehost/pdfviewer/pdfviewer?sid=cadf82db-4a91-42d3-9f12-eeea80f1203a%40sessionmgr12&vid=4&hid=28.This study aims to measure the effects of cooperative learning in secondary mathematics classes. The authors feel that although there is a substantial amount of research done on cooperative learning at the elementary age, there is little for secondary schools. Studies conducted with younger students found that cooperative learning increased social skills, positive affective results, and higher achievement. Whicker, Bol, and Nunnery expected that the same would hold true at the secondary level. They used a model similar to the Student Teams-Achievement Division, in which individual students earn points for their group based on the percent improvement on test grades. Students were 11th and 12th grade pre-calculus students in a rural high school. Students were told that they needed to explain their answers to each other and ask their group members for help rather than the teacher. The results indicated that students enjoyed working in groups; however, they did not like that groups were pre-assigned and permanent. Students found the groups most helpful when working on difficult concepts. Students in the experimental group scored better on chapter tests than students in the control group. Over time, this difference in scores increases. Overall, the study concludes that cooperative learning can be an effective instructional strategy in higher level, secondary mathematics courses. (K. Dwyer)

White, D. Y. (2003). Promoting productive mathematical classroom discourse with diverse students.

The Jounral of Mathematical Behavior,22(1), 37-53. http://dx.doi.org/10.1016/S0732-3123(03)00003-8Professor Dorothy White contrasts typical, direct teaching methodology with two classrooms that emphasize discourse in the classroom. White describes education in African American and Hispanic classrooms as geared toward repetitive problems meant to keep the class occupied and under control. The classes in the article portray an environment where the students thinking process is what is valued, far more than the correct answers. Namely, students were tasked with evaluating one another’s mathematical statements, rather than the teacher, and reasoning skills were applauded even when accompanied with an incorrect answer. In the classrooms described, students come to successful learning outcomes primarily by thinking through problems using their past experiences, participating in class discussion and the teacher’s prodding. White argues that this is evidence that African American and Hispanic children don’t need to be controlled, but need a venue for expressing their ideas, improving autonomous thinking, and attaining higher levels of understanding. (J. Traxler)

White-Clark, R., DiCarlo, M., & Gilchriest, N. (2008). Guide on the Side: An Instructional Approach to Meet Mathematics Standards.

High School Journal, 92(4),40-44.Typical high school students often express that they do not believe their mathematics courses are relating to their personal lives and feel disconnected to the instruction. Current research suggests a possible movement towards a more constructivist pedagogy exemplifying inquiry-based instruction. Constructivist pedagogy places a greater emphasis on student-centered instruction encouraging all students to become explorers in the content. Research has found that most teachers teach the way they were taught; therefore, teachers are still lecturing in a teacher-centered environment. What the article is trying to convey through the text is that teachers can influence student achievement when they alter traditional instruction. The authors offer teaching ideas about introducing mathematical concepts to students in various ways that requires students to explore with the mathematics and develop their own explanation of theories and practices. Through this kind of constructivist approach, the teachers should assume the role as a “guide on the side” allowing students to become actively engaged in the content and responsible for their learning. (S. Richards)

Williams, C. G. (1998). Using concept maps to assess conceptual knowledge of function.

Journal for Research in Mathematics Education 29(4), 414-421In this article, Williams discuses a study she conducted to “examine the value of concept maps as instruments for assessment of conceptual understanding” (p. 414). The study involved twenty-eight, first year calculus students and eight professors. Half of the students were enrolled in a traditional calculus course and half in a reform calculus course. The reform calculus course had an “emphasis on modeling and technology” (p. 414). The professors involved in the study held PhDs in mathematics. Four of the professors were from the same university as the twenty-eight students. The students and professors involved in the study attended an instructional session on concept maps. They were then given one hour to construct their own concept maps for “Function”. Williams then compared the concept maps of the three groups. Unlike the professors’ maps, she found the students’ maps to be widely divergent. Since the students had a “common understanding of the process for drawing a concept map, [this] wide diversity of maps derives mainly from participants’ different concepts of function” (p. 416). Many of the students’ maps also focused on algorithms. Williams concluded that there is “credibility to the conclusion that concept maps do capture a representative sample of conceptual knowledge” (p. 420). (L. Gainey)

Willig, C., Bresser, R., Melanese, K., Sphar, C., & Felux, C. (2009).

10 ways to help ELLs succeed in math. Http://www.scholastic.com/teachers/article/10-ways-help-ells-succeed-mathThis article stems from a book by the authors called:

Supporting English Language Learners in Math Class: A Multimedia Professional Learning Resource(Bresser et al., 2009). The article highlights ten strategies to help ELL students succeed in a mathematics class. The authors suggest that mathematics can be hard in any language. Therefore, they felt it was important to help other teachers by providing strategies. The following are the ten strategies: 1) Create Vocabulary Banks, 2) Use manipulatives, 3) Modify teacher talk and practice wait time, 4) Elicit nonverbal responses, like a thumbs up or down, 5) Use sentence frames, 6)Design questions and prompts for different proficiency levels, 7) Use prompts to support student responses, 8) Consider language and math skills when grouping students, 9) Utilize partner talk, and 10) Ask for choral responses from students. Besides the 10 strategies, the authors also gave tips specifically for teachers. Examples are also provided for the strategies. (R. McDowell)

Wong, B., & Bukalov, L. (2013). Improving Student Reasoning in Geometry.

Mathematics Teacher, 107(1),54-60.Wong and Bukalov feel that there is a need for improvement in students’ reasoning skills in geometry classes. They feel that the best strategy may be tiering lessons. The tiered lessons consist of four levels where level 1 is the simplest and level four is the most challenging. Having multiple levels allows students to choose their level of understanding and starting point. Presenting problems in this manner allows access to the problems while still challenging them to learn more. The sample activity presented demonstrates the use of practice problems as well as guided questions to help students advance levels. By creating a tiered lesson, students with varying degrees of understanding can work on the same subject matter. It also allows for the entire class to participate in discussions at the beginning and end of class without anyone being left out. The idea of a tiered lesson is a very interesting way of differentiating instruction while allowing students to choose a level suited to their own abilities. The challenges of such lessons are also presented. After having read this article, I would consider activities like the one discussed on occasion. (K. Dwyer)

Wood, J. (2013). Using aviation to change math attitudes.

Mathematics Teaching in the Middle School, 18(7), 408-414. http://www.nctm.org/publications/article.aspx?id=35639.In this article Jerra Wood discusses a lesson from the Kentucky Aviation Teacher Institute (KATI) that Tim Smith of the Kentucky Institute of Aerospace Education developed. The KATI believes that “teaching mathematics within the context of aerospace generates excitement and interest among students” (p. 410). The lesson discussed in the article introduces linear equations using the Microsoft Flight Simulator. Students used the simulator to pilot a plane and collect data during takeoff. However, before the students could begin, they had to learn how to read the air speed indicator, the altimeter, and the attitude indicator. This lesson used a real-world context to introduce a new topic. It also used the same technologies professionals use out in the field. Using real-world context problems, like the one presented in this lesson, can show students the relevance of the mathematics they are learning and provide motivation. (L. Gainey)

Yerushalmy, M., & Gilead, S. (1997). Solving equations in a technological environment.

Mathematics Teacher 90(2), 156-153.

http://www.nctm.org/publications/article.aspx?id=17785The authors open with the question, “Once symbolic manipulations are disconnected from the process of reaching a solution, what is the merit of carrying out such manipulations” (p. 268)? The authors also ask if the use of technology alone is enough to convey meaningful mathematics. To answer these questions, they tested an alterative algebra course in Israel that incorporated technology. The students in this course were assessed using tasks that included both familiar and novel problems. In the article, the authors discussed the solutions several students provided on two of these tasks. In addition to discussing student solutions, they discussed how they took student solutions from the previous unit’s assessment into consideration when deciding how to approach the following unit. Going back to the questions posed at the beginning of the article, the authors concluded that technology alone is not sufficient, but that “the nature of the tasks, which the technology made interesting to explore, changed the form of the study of manipulations” (p. 274). (L. Gainey)