EMAT 7050 Annotated Bibliography

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education.  Elementary School Journal, 90 (4) ,449–466.

Deborah 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)

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)

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)    

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)

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)

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)

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)

Larson, M. R., & Leinwand, S. (2013). Prepare for More Realistic Test Results.Mathematics Teacher, 106 (9), 656-659.

This 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)

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)

Moschkovich, J. (2009). How can research help us understand mathematics learners who use two languages?Research Brief , National Council of Teachers of Mathematics.

This 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)

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)


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)

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

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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.

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)

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)

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)

Glasersfeld, E. von (1981). Einführung in den radikalen konstrucktivismus. In P. Watzlawick (ed.) Die erfundene Wirklichkeit. Munich: Piper, pp. 16–38. English translation: An introduction to radical constructivism. In P. Watzlawick (ed.) (1984) The invented reality. New York: Norton, pp. 17–40.

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)

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)

Martin, D. (2003). Hidden assumptions and unaddressed questions inMathematics for Allrhetoric.The Mathematics Educator, 13 (2), 7-21.

This 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 gettingtherethan 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.

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 theMathematics 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)

Scher, D. (1997). Dynamic visualization and proof: A new approach to a classic problem.Mathematics Teacher 96 (6), 394-398

In 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)

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.)

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 is published 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).

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

White, D. Y. (2003). Promoting productive mathematical classroom discourse with diverse students. The Jounral of Mathematical Behavior ,22 (1), 37-53.

Professor 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)

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