EMAT 7050 Mathematics Instruction

Fall 2014


August 27 Entries
September 3 Entries
September 10 Entries
September 17 Entries
September 24 Entries
October 1 Entries
October 8 Entries

October 22 Entries
October 29 Entries
November 5 Entries
November 12 Entries
November 19 Entries


August 27 Entries

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)

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)

Jackson, K., Garrison, A., Wilson, J., Gibbons, L., & Shahan E. (July 2013). Exploring relationships between setting up complex tasks and opportunities to learn in concluding whole-class discussions in middle-grades mathematics instruction. Journal for research in mathematics education, 44(4), 646-682.

Many modern teachers use a lesson model that includes a cognitively demanding task, followed by a class discussion of the students’ results. This article investigates the effects that a task’s introduction can have on student learning. The authors examined instruction by 165 middle-school teachers for key features related to a lesson’s set-up and discussion. Results showed that teachers who emphasized the mathematical relationships prior to beginning a task tend to have more academically rigorous discussions. Fewer teachers explained the contextual features of a problem, but their students were more likely to make connections between each other’s ideas. One troubling statistic is that about 64% of lessons showed a lowered level of cognitive demand throughout the class. The conclusions from this study are helpful for teachers to remember when introducing a task to students. (B. Roper)

Knuth, E.J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Reseach in Mathematics Education, 33(5), 379-405.

Knuth interviews 16 in-service secondary school mathematics teachers to identity their conceptions of what constitutes proof and what is convincing to them as proof. The theoretical perspective used to frame the research is based on that of Hanna’s (1990) work on proof. Surprisingly, Knuth found that several teachers believed proofs to be able to be proven false, even if they are a valid proof. Teachers also tended to omit understanding as a purpose of proof. It was clear that some teachers’ perceptions of proof do not match those of mathematicians and what is convincing as proof tends rely on empirical evidence as opposed to “mathematical substance of the argument” (p. 402). Knuth concludes that 1) college mathematics teachers and teacher educators are in part to blame for not adequately preparing teachers in proof and 2) more work needs to be done to determine how this view affects their teaching of mathematics. (J. Przybyla-Kuchek)

Mavrikis, M., Noss, R., Hoyles, C., & Geraniou, E. (2012). Sowing the seeds of algebraic generalization: designing epistemic affordances for an intelligent microworld. Journal of Computer Assisted Learning, 29, 68-84.

This is an article about designing of teaching environment to help secondary school students motivate and exploit algebraic ways of thinking, which is a core stumbling block in secondary school mathematics classrooms. In order to deal with these challenges, the authors undertook an epistemological analysis of algebraic generalization and designed a computer environment that addresses its key components, which consists of perceiving and exploiting structure, seeing the general in the particular and expressing generalizations symbolically. 12 to 13-year-old students participated in this project and interacted with computer based visual pattern making activities, called the eXpresser. In conclusion, the authors could see this intelligent activities’ potential to support the development of algebraic ways of thinking. (D. Shin)


Morgan, A., Parr, B., & Fuhrman, N. (2011). Enhancing Collaboration among Math and Career     and Technical Education Teachers: Is Technology the Answer? Journal of Career and Technical Education, 26(2), 77-89. (2011, December 1). Retrieved August 26, 2014,  from ERIC, http://eric.ed.gov/?id=EJ974466.

For this study researchers took a convenience survey of 44 teacher educators after attending a seminar on collaborative learning across Mathematics and Career and Technical Education (CTE). The study consisted of both qualitative questions that were opened ended as well as quantitative responses. The purpose of this study was to determine the perceptions teachers had about the “value” and “willingness to implement” practices and activities suggested at the seminar in the classroom, in addition to determining the pros and cons of technology as a means of collaboration. The results in the quantitative portion of the survey indicated that teachers placed a high value on integrating the Math and CTE curriculum. However, in the qualitative responses, teachers noted that time and administrative support were both significant barriers to the implementation of cross curriculum material. Teachers were overall very cautious about the new ideas, and hoped to discuss it further with other colleagues before regularly integrating it into their own classroom. (S. Erwin)

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, Heuristic is a field of study and he descibes How to Solve It as 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)

Riegle-Crumb, C. & Grodsky, E. (2010). Racial-Ethnic Differences at the Intersection of Math Course-taking and Achievement. Sociology of Education, 83 (3), 248-270.

Although strides have been taken to increase equity in mathematics classes, there is a clear achievement gap between Hispanic and African American students and white students. The gap is evident in all levels of mathematics and becomes larger in more advanced high school classes. In general, the number of students taking advanced level classes is not proportional to the population of the school. In schools that are predominantly African American, there is a wider gap that is closing more slowly. Interestingly, minority students typically attend schools that have greater than 50% population of minority students, while the majority of white students attend schools that are approximately 15% minority. This post-modern segregation is one factor of the growing achievement gap and differences in course-taking. (K. Patel)


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)

Webel, C. (February 2013). Classroom Collaboration: Moving Beyond Helping. Mathematics Teacher, 106(6), 464-467. Retrieved from http://www.nctm.org/publications/article.aspx?id=35301

This article investigates the many components that come with group work.  Author, Corey Webel, observed a high school class where students often work collectively solving open ended problems and throughout his research, interviewed a handful of students as part of his investigation.  He presents two contrasting views that come with group work, asymmetric positioning and symmetric positioning, and gives details about three particular students who have shown one or the other.  Asymmetric positioning is when students think that “getting help” means that there are some students who will be considered experts on the topic or problem and the other students, being called the novices, get help from them.  Behaviors among students who think the whole group is responsible, work to resolve disagreements, and collect ideas together is called symmetric positioning.  Webel presents the problem the class was given to work on, identifies which students are considered asymmetric and symmetric, gives a brief report on how the groups worked on the problem, and then presents the individual interviews with those students.  This article is a great insight into different ways that group work played out in a mathematics classroom.  (B. Koblitz)

September 3 Entries

Clarke, L.M., DePiper, J.N., Frank, T.J., Nishio, M., Campbell, P.F., Smith, T.M., Griffin, M.J., … Choi, Y. (2014). Teacher Characteristics Associated with Mathematics Teachers’ Beliefs and Awareness of Their Students’ Mathematical Dispositions. Journal for Research in Mathematics Education, 45 (2), 246-284.


The article highlights a relationship between being aware of students’ dispositions in math, belief that students should be allowed to struggle,  belief that teachers should model for incremental mastery, and teachers’ educational background. Allowing students to struggle is related directly tied to teacher understanding of the mathematics. The study looked at how teacher education influences these beliefs and how they affect the achievement of students. Personal experiences of teachers were addressed as having a significant effect on beliefs in addition to education. It was suggested that teacher education programs need to add explicit ways for teacher candidates to formulate their beliefs. (K. Patel)

DeJarnette, A. F., Dao, J. N., & Gonzalez, G. (March 2014). Learning What Works: Promoting Small Group Discussions. Mathematics Teaching in the Middle School, 19(7), 414-419.  Retrieved from http://www.nctm.org/publications/article.aspx?id=41260

This article talks about middle school students who are put into groups to work on a rate of change problem.  In this particular classroom, students had worked in groups together before but never on a problem that would take the whole class period.  The researchers reported that the behavior of the groups were collaborative and productive.  From observing the groups, they came up with three main strategies that can promote productive mathematical discussion amongst students working in groups: asking questions about the problem, sharing the mathematical authority in the group, and challenging each other’s mathematical ideas.  They provide interesting insight as to why students may struggle at first with asking questions to each other because they are usually used to the teacher initiating discussions and questions.  Through interactions presented with different groups, the researchers showed how one strategy leading to the next can fix the things that were holding groups back from having a fully productive session.  A table of ideas is also presented in the article of things that teachers can do to support the collaboration of students. (B. Koblitz) 

Knuth, E. J., Stephens, A. C., McNeil, N .M., & Alibali, M. W. (July 2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for research in mathematics education, 37(4), 297-312.

Middle school algebra students use the equal sign nearly every day, but many of them do not fully understand its meaning. Rather than representing equivalence, they believe that the symbol indicates an operation or answer. The authors examined an assessment taken by 117 middle school students that asked them to interpret the equal sign and solve various algebraic equations. Less than half of students in each grade gave a relational response, and the percentage did not increase for each successive grade level. The research also concluded that there is a correlation between understanding the equal sign’s meaning and success in solving equations. This article demonstrates the importance of further developing the meaning of the equal sign throughout students’ middle grades education. (B. Roper).

Mgombelo, J., & Buteau, C. (2012). Learning Mathematics Needed for Teaching Through Designing, Implementing and Testing Learning Objects. Issues in the Undergraduate Mathematics Preparation of School Teachers, 3. Retrieved September 1, 2014, from ERIC, http://files.eric.ed.gov/fulltext/EJ990481.pdf

            This journal article is an overview of a study on pre-service teachers conducted at Brock University in Canada. The purpose of the study was to explore how designing interactive computer modules called “Learning Objects” influenced the pre-service teachers’ understanding of mathematics. Learning Objects are part of an overall program called Math Integrated with Computers and Applications (MICA) where pre-service teachers are taught how to teach mathematics by integrating it into computer programming and technology. The researchers were motivated to do this study because they found that although teachers knew the mathematics they would later teach, they did not know how to effectively explain and teach their knowledge. The students who participated in this study were required to create their own working Learning Object, and the researchers found that through this process, students were much more engaged in the mathematics and were better equipped to teach the content. (S. Erwin)

Piatek-Jimenez, K. (2008). Images of mathematicians: a new perspective on the shortage of women in mathematical careers. ZDM, 40(4), 633-646.

What beliefs do you hold about mathematicians? Do you think of yourself as a mathematician? Piaket-Jimenez asks these questions along with several others in a an attempt to identify why women who choose mathematics as a major in college do not pursue a career in mathematics or continue on to graduate school in mathematics. Although several other studies have searched for this answer, Piatek-Jimenez uses the theoretical lens of identity to incorporate beliefs about mathematicians into the complexity of social influences involved with choosing careers. Her results show that the five women she interviewed held the following beliefs about mathematicians: “(a) mathematicians are extremely intelligent, (b) mathematicians are obsessed with their work in mathematics, and (c) mathematicians do not fit in with social norms” (p. 641). All the women in this study could not identify with at least one of the attributes, contributing to their perceptions of themselves as mathematician-in-training and possibly their decisions to pursue a career in mathematics. What implications could this have for secondary education? (J. Przybyla-Kuchek)

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 case study examines the teaching style of a tenth grade geometry teacher who based on classroom management style, student achievement on state standardized assessments, and classroom interactions with students would be considered an effective teacher. Upon closer examination, Schoenfeld is able to show that the teaching style employed by the target teacher reinforces certain behaviors and habits that negatively impact their mathematical performance and belief systems related to problem solving. (K. Keels)

White, T. (2013). Networked Technologies for Fostering Novel Forms of Student Interaction in High School Mathematics Classrooms. In Emerging Technologies for the Classroom (pp. 81-92). Springer New York.

This article examines novel forms of mathematics teaching, learning, and classroom interaction supported by local networks of handheld calculators and computers. The author presents three different interactive classroom activities that use classroom networks, each drawn from research projects focused on investigating the potentials of these tools for supporting novel forms of teaching and learning mathematics. The activities are consisted of small-group collaboration (among students) and whole-group activity structures (between students and teacher) in order to facilitate student participation. The potentials through these approaches are that the small- and whole-group activities can serve as important exemplars for teachers and designers seeking to capitalize on classroom networks as resources for supporting students’ interactions with one another and with important ideas in mathematics. (D. Shin)

Xin, Y. P. (2007). Word Problem Solving Tasks in Textbooks and Their Relation to Student Performance. The Journal of Educational Research, 100(6), 347-359.

The intended curriculum in a classroom comes from the textbook, but are all textbooks created equal especially in regards to how they present word problems?  Xin did a cross-cultural study to determine the influence that textbooks had on problem-solving abilities for students with learning disabilities in the United States and China. Several problem types involving multiplication and division were identified in the study, and the textbooks used by the participants were analyzed to determine the frequency that each problem type appeared. Students were also given a word problem assessment that only focused on multiplication and division concepts. While interpreting the data collected from the student assessments and textbook analysis, it was apparent that the Chinese textbooks offered more of a balance in the representation of the word problem types identified in the study, which positively impacted the problem solving assessment results for the Chinese students.  (K. Keels)

September 10, 2013

Abedi, J., & Lord, C. (2001). The language factor in mathematics tests. Applied Measurement in Education, 14(3), 219-234.

This article reports on research conducted to determine if the language of a mathematics test -- whether complex or simplified -- can affect students’ performance, particularly those with low English proficiency. The article explains how the research is conducted and finds that indeed, students performed better on the linguistically more simplified problems. This article is a testament to the fact that even if a student understands the mathematics well, he or she can still do poorly on an exam, solely due to a language barrier. Many people forget to think of language as being important in mathematics, but this article shows that it is highly important indeed. (S. Erwin)

Bryant, D. P., Bryant, B. R., Gersten, R., Scammacca, N., & Chavez, M. M. (2008). Mathematics intervention for first- and second-grade students with mathematics difficulties: The effects of tier 2 intervention delivered as booster lessons. Remedial and Special Education, 29(1), 20-32.

The purpose of this study is to determine the effectiveness of Tier 2 intervention on first and second grade students who are considered struggling learners in mathematics. Several number sense concepts that struggling learners experience difficulty with were identified within this study, and intervention was developed to support the development of these skills. Supplemental tutoring sessions related to number sense were offered several times per week over the course of 18 weeks. These booster lessons resulted in no significant change in the first grade students, but there was a significantly positive effect on the second grade math students. (K. Keels)

Dick, T. (2008). Keeping the faith. Fidelity in technological tools for mathematics education. In G. W. Blume & M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics: syntheses, cases, and perspectives. Vol. 2: Cases and perspectives (pp. 333–339). Greenwich, CT: Information Age Publishing.

There are lots of mathematic education technologies such as computer programs, calculators and so on. But how well are these technological tools reflecting user’s cognitive actions, mathematical accuracy, and pedagogical faith? In the article, Thomas P. Dick discusses about fidelity. In detail, it means that technology designed for use in mathematics education should be faithful to some basic principles. He refers to these principles as pedagogical fidelity, mathematical fidelity, and cognitive fidelity. He suggests how can we efficiently move from the idea for a mathematics learning activity to the implementation of a technological tool that facilitates that activity and remains faithful to it, focusing on the fidelity principles. (D. Shin)

Gaddy, A.K., Harmon, S.E., Barlow, A.T., Milligan, C.D., Huang, R. (September 2014). Implementing the common core: applying shifts to instruction. Mathematics Teacher. 108(2), 108-113.

The authors identify three key areas where teachers will need to adjust their instruction for the Common Core Standards for Mathematics: focus, coherence, and rigor. A brief explanation of how these design principles influenced the creation of the standards is provided. Teachers should now focus on conceptual understanding rather than procedures, maintain the coherence of concepts through difference grade levels, and provide rigorous tasks with applications of mathematical ideas. These disciplines are illustrated through a classroom example of students extending their knowledge of linear functions to arithmetic sequences. Transcripts of conversations and samples of student work are included as the class works on a task involving a pet hotel. The whole-class discussion and multiple solution strategies of the lesson demonstrate the changes that the authors claim are required by the new standards. (B. Roper)

Linsenmeier, K.A., Sherin, M., Walkoe, J., & Mulligan, M. (2014). Lenses for Examining Students’ Mathematical Thinking. The Mathematics Teacher, 108(2), 142-146.

This article is from The Mathematics Teacher and is part of the connecting research to practice series. It focuses on three lenses through which a teacher can investigate student thinking. The article gives specific steps for teachers to implement the strategies and questions to ask while working with students. A professional learning video club is the basis for these conversations and the article talks a little bit about how to implement a video club. However, the cited sources go into this much deeper. (K. Patel)


Mendick, H. (2005). Mathematical stories: why do more boys than girls choose to study mathematics at AS-level in England? British Journal of Sociology of Education, 26(2), 235-251.

With respect to the article by Piatek-Jimenez (2008), Mendick’s article seeks answers to a similar question but from a different framework and in a different culture. This allows the reader to compare and contrast the reasons young adults choose (or do not choose) to study mathematics at a more advanced level through the lens of gender. Medick’s framework is rooted in post-structuralist theory with an emphasis on power. She tells the stories of female two students (of the 43 total participants in the study) and tensions they have between the masculinity in mathematics and identifying themselves as females. Mendick concludes that “seeing ‘doing mathematics’ as ‘doing masculinity’ is a productive way of understanding why mathematics is so male dominated” (p. 235). (J. Przybyla-Kuchek)

Stein, M. K. (October 2001). Mathematical Argumentation: Putting Umph into Classroom Discussions. Mathematics Teaching in the Middle School, vol. 7 (no. 2). Retrieved from http://www.nctm.org/publications/article.aspx?id=19934

In this article, Mary Kay Stein discusses how NCTM Standards recommend to orchestrate classroom discourse, but that this is one of the most difficult things for a teacher to do.  The first part of provoking classroom discussion is a good task.  Through research, Stein has found that choosing the right tasks and with coaching from the teacher, the role of teacher centered classrooms can be broken.  She even believes that she has shown proof through research that healthy mathematical argumentation can even be accomplished in low-income, urban neighborhoods.  The summary of her research that is presented in this article is very insightful for the benefits of teachers pushing students to discuss mathematics with each other. (B. Koblitz)



September 17, 2013

Abramovich, S., Schunn, C., & Higashi, R. (2013). Are badges useful in education?: It depends upon the type of badge and expertise of learner? Educational Technology Research and Development, 61(2), 217-232. (2013, April 1). Retrieved September 15, 2014, from ERIC.

Extrinsic motivators are a controversial subject in education. On the one hand, most students will do something for candy or a small treat; however, at what point does it become harmful to the student? This study examines extrinsic rewards, called “badges,” and their effect on middle school students. A computer program rewarded either skill or participatory based badges accordingly to students. The researchers issued a pre-test and post-test of student performance and motivation, trying to determine a correlation between motivation, performance, and badges. Indeed, they found that for low-level students, earning participatory badges did decrease their worry about performance, but earning more related to less of a decrease. For high-performing students, earning skill badges related to an increase in expectancy to do well. (S Erwin)

Brown, M., Brown, P. & Bibby, T. (2008). “I would rather die”: Reasons given by 16-year-olds of not continuing their study of mathematics. Research in Mathematics Education, 10(1), 3-18.

The authors investigate the reasons 16-year-olds opt out of continuing mathematics at the AS- and A-levels in England. In contrast to the previous studies presented in class (Piatek-Jimenez, 2008 & Mendick, 2005), this is a quantitative study with a large sample of 1510 students from 17 schools and a goal of determining why students are NOT continuing with mathematics, as opposed to determining why students choose to continue mathematics. The authors analyze the data from a short survey with respect to students predicted grade, gender, and attending school. Although the study does not bring about significantly startling results, the results are interesting none-the-less. In addition, the authors are able to confirm results from previous studies and provide suggestions for encouraging more students to study maths. (J. Przybyla-Kuchek)

Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 352-378.

This study supports previous findings regarding the complexity of reasoning about covarying relationships. However, the results of this study extend what has previously been reported by identifying specific aspects of covariational reasoning that appear to be problematic for college-level students. In order words, even though the subjects of this study are high-performing undergraduate students, they appeared to have difficulty constructing images of continuously changing rate and could not accurately represent or interpret inflection points or increasing and decreasing rate for dynamic function situations. From these observations and previous studies, the authors propose a framework for describing the mental actions involved in applying covariational reasoning when interpreting and representing dynamic function events. They hope that the study’s results and the covariation framework will serve to explicate the cognitive actions involved in students’ reasoning in interpreting and representing dynamic function events. (D. Shin)

Jitendra, A. K., Rodriguez, M., Kanive, R., Huang, J., Church, C., Corroy, K. A., & Zaslofsky, A. (2013). Impact of small-group tutoring interventions on themathematical problem solving and achievement of third-grade students with mathematics difficulties. Learning disability quarterly, 36(1), 21-35.

This study compares the overall effects of two different small-group tutoring treatments. Each treatment focused on fostering the development of problem solving skills in third-grade students who experience math difficulties. Tutors were trained in either schema-based instruction or the standards-based curriculum used in the school setting. Each participant completed a pretest and a posttest as a part of the data collection process, and those test scores were analyzed in order to determine the efficacy of each type of small-group instruction.​ (K. Keels)


Lim, K.H. (2014). Error-eliciting problems: fostering understanding and thinking. Mathematics Teaching in the Middle School. 20(2), 106-114.

Rather than simply being unexplored failures, Lim argues that mathematics teachers can use student errors as opportunities for further discussions. These learning extensions involve three of the eight Standards for Mathematical Practices. The ability to control the mistakes in a classroom depends on whether they are planned, expected, or unexpected. Lim provides two examples each for problems that elicit three types of errors: misconceptions, misapplications, and overgeneralizations of a concept. For each item, he explains the reasoning of a student who made that error. The article also includes statistics on how many prospective teachers committed the expected errors. Each of these questions can help students realize their procedural impulses and consequently, analyze a situation more closely. Finally, Lim gives several suggestions for creating and implementing one’s own error-eliciting tasks. (B. Roper)

Lobato, J., Hohensee, C., & Rhodehamel, B. (2013). Students’ Mathematical Noticing. Journal for Research in Mathematics Education, 44(5), 809-850. http://www.jstor.org.proxy-remote.galib.uga.edu/stable/10.5951/jresematheduc.44.5.0809

Students notice different things as they investigate mathematics. This article focuses on the effect of what students notice on their mathematical reasoning through tasks that investigate linear relationships. In the study, two classes approached understanding linear relationships in two different ways, or through different centers of focus. Each class also had different focusing interactions- highlighting, quantitative dialogue, and renaming. These focusing interactions led students to notice and value certain mathematics over others. Students’ noticing had effects on their reasoning in future problems. The research showed that teachers have a critical role in directing students’ attention toward or (often unintentionally) away from what is centrally important. There are subtle moves that teachers make that can have a large affect on what students notice mathematically. (K. Patel)

Mulryan, C. M. (1994). Perceptions of Intermediate Students’ Cooperative Small-Group Work in Mathematics. The Journal of Educational Research, 87(5), 280-291.
Retrieved from http://www.jstor.org.proxy-remote.galib.uga.edu/stable/pdfplus/27541931.pdf?&acceptTC=true&jpdConfirm=true

This article discusses research showing that help given and received in small group work can improve achievement in mathematics classrooms.  There is a focus on teachers’ and students’ perceptions that help give information on instruction and making group work cooperative.  Mulryan uses a more rare study of perceptions of high and low achieving students, as well as male versus female students, to gain an understanding for further research in the field of cooperative group work.  I chose this article because I like that it compares the perceptions on different parts of group work between teachers and students.  It also breaks down some of the research nicely into different tables. (B. Koblitz)  



September 24 Entries

Dekker, R. & Elshout-Mohr, M. (2004). Teacher Interventions Aimed at Mathematical Level Raising during Collaborative Learning. Educational Studies in Mathematics, 56(1), 39-65.

This article investigates how different types of help provided to students working collaboratively in groups can affect their understanding of a mathematical problem and raise their understanding to a higher level.  The mathematical levels that are mentioned refer to Van Hiele’s three different levels of mathematical understanding and competence.  The two kinds of help that are used are process help and product help.  Process help is more focused on student interaction and the social aspect rather than mathematical problem solving of the group, while product help is concerned with the students’ mathematical reasoning and products.  In this particular experiment, there were two groups of students ages 16-17selected and one group was given process help, while the other was given product help.  They took a pre test and post test and a task from which the results showed that the students exposed to process help conditions raised their mathematical level more than product help. (B. Koblitz)

Abrahamson, D. (2012). Seeing chance: perceptual reasoning as an epistemic resource for grounding compound event spacesZDM44(7), 869-881.

In this paper the author addresses one of the challenges in education: the content of random compound events. Conventionally, educators have addressed this conceptual difficulty by engaging students in actual experiments whose outcomes contradict the erroneous predictions. Even though empirical activities are important for any probability design, the author proposes to consider perceptual reasoning as an alternative or complementary epistemic resource for students to understand compound events. He supports his view by detailing literatures as well as by presenting empirical findings from a design-based research project. This research demonstrates that under auspicious design and steering, naïve perceptual reasoning can lead students to accept the differentiation of variations. (D. Shin)

Laursen, S.L., Hassi, M. L., Kogan, M. & Weston, T. J. (2014). Benefits for women and men of inquiry-based learning in college mathematics: A multi-institutional study. Journal for Research in Mathematics Education, 45(4), 406-418.

This is a brief report of a large study conducted in four universities, including over 100 course sections. Although there are several articles reporting on different aspects of the study, this particular article reports on the benefits for both men and women in Inquiry-Based Learning (IBL) courses. The authors compared IBL courses to lecture based courses using a quasi-experimental design. Students in both types of courses were given pre- and post-tests designed to reflect students’ beliefs, attitudes and approaches to learning mathematics. Data was also collected on students’ perceived cognitive, affective, and collaborative gains. Results show that both genders could gain from IBL, but women in particular showed more gains in attitudes toward and confidence in mathematics. (J. Przybyla-Kuchek)

Lewis, K. E. (2014). Difference not deficit: Reconceptualizing mathematical learning disabilities. Journal for research in mathematics education, 45(3), 351-396.

Traditionally, students with mathematical learning disabilities (MLD) have been identified based on deficits and lack of understanding. Lewis, however, suggests that this focus should be shifted to detecting differences between typical mathematics students and students who experience MLD. Using Vygotskian principles to redefine mathematics disabilities, she discusses several persistent understandings that students with mathematics disabilities develop during a fraction intervention unit. She claims the traditional representations used to support learning in a classroom may be inaccessible to these students. Further research on the different learning strategies used by students with MLD may help develop meditational tools that are both accessible and compatible with these students’ cognitive processes. ​(K. Keels)

Sullivan, P., Tobias, S., & McDonough, A. (2006). Perhaps the Decision of Some Students Not to Engage in Learning Mathematics in School Is DeliberateEducational Studies in Mathematics, 62(1), 81-99. Retrieved September 23, 2014, from ERIC.

This is a study of 13 year old students in Mathematics in four classes at a middle school in Australia. The researchers wanted to know what motivated the students to do well in mathematics, and how they would persevere through a difficult task. It is interesting to note that across the board, all levels of achievement of students persevered in the mathematics task given. However, only the high level students indicated that math is useful outside of the classroom. This study gives teachers an idea of how to modify the classroom environment so that the students are more motivated, including providing the useful of math in the real world, working one on one with students, and creating an environment where students will not be teased, whether excelling or struggling in mathematics. (S. Erwin)

Walkington, C., Sherman, M., Petrosino, A. (2014). “Playing the game” of story problems: coordinating situation-based reasoning with algebraic representation. Journal of Mathematical Behavior. 31(2). 174-195.

The study presented in this article explores whether a context can provide students with better access to mathematical ideas. This research differs from previous work in that it focuses on high school students and algebraic word problems, rather than elementary school students and arithmetic word problems. After a selection process, 24 students in an Algebra 1 class from an urban high school in Texas participated in a problem-solving interview. They were asked to complete four to five types of problems that were personalized for each student from their views on the usefulness of mathematics. After the interviews, researchers identified three primary processes that impair or assist students’ understanding of the problems: verbal and symbolic representation, use of situation-based reasoning, and coordinative and non-coordinative problem-solving approaches. Examples of student work and transcripts are provided to illustrate each of these behaviors. (B.  Roper) 



October 1 Entries

Henningsen, M. & Stein, M. K. (November 1997). Mathematical Tasks and Student Cognition: Classroom-Based Factors That Support and Inhibit High-Level Mathematical Thinking and Reasoning. Journal for Research in Mathematics Education, 28(5), 524-549. Retrieved from http://www.jstor.org.proxy-remote.galib.uga.edu/stable/pdfplus/749690.pdf?acceptTC=true&jpdConfirm=true

In this article, Henningsen and Stein focus on students’ engagement in mathematical tasks, and more specifically high cognitive demand tasks.  A study is conducted to see what types of different factors within a classroom can affect if students are engaged or not.  Results are presented in four different bar graphs with one showing factors that maintained engagement and high cognitive demands, and the other three breaking down the three different types of decline in engagement and their factors.  These four focuses are also presented with specific classroom observations and breakdown of the tasks. (B. Koblitz)

Irwin, K.C. (2001). Using everyday knowledge of decimals to enhance understanding. Journal for Research in Mathematics Education. 32(4). 399-420.

This study examines whether New Zealand students in a lower socioeconomic area and minority culture can improve their understanding of decimal concepts by solving contextual problems. Previous research on the subject has shown that these students can be more resistant to learning new mathematical ideas through everyday scenarios. 16 students took a pretest over decimal representations and operations, followed by a three-day intervention. They were divided into two groups: one was given problems with a context and the other without one. In the groups, higher achieving students were paired with lower ones to also investigate peer collaboration. After a posttest, there was a greater increase in the scores of the students under the contextualized intervention than the other type. By analyzing the student dialogues, Irwin also found that the higher ranked students often ignored the contexts and spent more time manipulating the numbers than their partners. (B. Roper)

Izsak, A., Tillema, E., and Tunc-Pekkan, Z. (2008). Teaching and Learning Fraction Addition on Number Lines. Journal for Research in Mathematics Education, 39(1), 33-62. http://www.jstor.org.proxy-remote.galib.uga.edu/stable/30034887

This article is a case study of a class of 6th grade students who are learning about fraction addition using number lines as a tool for understanding. I chose this article because the curriculum my school uses is the Connected Mathematics Project which is what the researchers used as classroom materials. Through the study, researchers focused on how students interpreted the information and how students interpreted the information. The differences show how student perception of mathematical tasks varies, especially in reference to things that teachers believe to be subtle. (K. Patel)

Powell, S. R. & Fuchs, L. S. (2010). Contribution of equal-sign instruction beyond word problem tutoring for third-grade students with mathematics difficulty. Journal of Educational Psychology, 102(2), 381 – 394.

Misinterpreting the equal sign is a common issue among elementary students. Instead of treating the equal sign as a relational symbol, they view it as an operational symbol that indicates a specific calculation needs to be found in order to reach a solution. In this study, the author evaluates the efficacy of small group instruction focused on equal-sign concepts when combined with word problem intervention for third-grade students with mathematics difficulty. Participants in this study either received a combination of equal-sign and word problem intervention, word problem intervention-only, or no intervention at all. When comparing these three conditions, students who received a combination of equal-sign instruction and word problem intervention significantly outperformed their counterparts when solving open equations and certain forms of word problem tasks. (K. Keels)

Stage, F. K. & Maple, S.A. (1996). Incompatible goals: Narratives of graduate women in the mathematics pipline. American Educational Research Journal, 33(1), 23-51.

Although this is an older article, the research questions addressed here are still being investigated (Piatek-Jimenez, 2008; Mendick, 2005, Brown, Brown, & Bibby, 2008): Why do women enter and why do women leave the mathematics pipeline?  Stage and Maple interviewed seven white, middle class, American-born women enrolled in doctoral programs to obtain narratives about their decisions to obtain bachelors’ degrees in mathematics and eventually leave a mathematics graduate program, their relationship with mathematics, and the relationship between the perceptions of mathematics and perceptions of themselves. Several themes emerged from the data, some of which are: a) interest in mathematics early on; b) the influence of a role model (teacher/parent); c) the perception of mathematics as challenging or game-like; and d) specific negative experiences with mathematics leading to their decisions to leave. Finally, recommendations are given to help retain female students in graduate level mathematics programs. (J. Przybyla-Kuchek)

Xu, F., & Denison, S. (2009). Statistical inference and sensitivity to sampling in 11-month-old infantsCognition, 112(1), 97-104.

In this article, the authors investigate whether infants are sensitive to sampling conditions and whether they can integrate intentional information in a statistical inference task. 72 11-month-old infants participated and three experiment conditions were included in this study: a random sampling condition, a non-random sampling condition, and a blindfold condition. The infants were randomly assigned to each of three conditions. The results of this experiment showed that infants were sensitive to whether a sample had been drawn randomly from a population or not. Even more impressively, the infants were able to integrate multiple sources of information in deciding whether to employ the statistical inference mechanism. This indicates that infants could override the random sampling assumption, but only under appropriate circumstances. (D. Shin)

Xu, F. & Garcia, V. (2006). Intuitive Statistics by 8-Month-Old InfantsProceedings of the National Academy of Sciences, 105(13), 5012-5015. (2008, April 1).

This is a report on multiple studies done by researchers with infants and intuitive statistics. The researchers wanted to determine if 8-month-old infants could make predictions of a population based upon a sample, and if these children could make predictions of a sample based on a population. After conducting six experiments in total, they determined that 8-month-old infants were indeed capable of using information in a sample to make inferences about a larger population, as well as using a population to determine a viable sample. They use these findings to make the argument that humans are rational learners early in development. (S. Erwin)





October 8 Entries

Boaler, J. & Staples, M. (2008). Creating mathematical futures through an equitable teaching approach: The case of Railside school. Teachers College Record, 100(3), 608-645.

Jo Boaler and her research team conducted a 4-year longitudinal study in three high schools (two suburban and one urban) with 700 participants in California. Their goal was to better understand equitable and successful teaching of mathematics. Qualitative and quantitative methods were utilized to compare achievement and attitudes of students, curriculum approaches, and practices/norms of the classrooms among the three schools. This study focuses on the community of teachers and students at one of the schools, Railside, where a reform-oriented approach to mathematics was implimented. Results from Railside showed that “mathematical materials and associated teaching practices that encouraged students to work in many different ways, supporting the contributions of all students, not only resulted in high and equitable attainment, but promoted respect and sensitivity among students”(p. 640). It is important that there was no one contributing factor to the successes at Railside. Instead, it was a complex integration of curriculum, strong beliefs, dedicated teachers, and mixed-ability classrooms. (J. Przybyla-Kuchek)

Hyde, J.S., Else-Quest, N.M., Alibali, M.W., Knuth, E., & Romberg, T. (2006). Mathematics in the home: Homework practices and mother-child interactions doing mathematics. Journal of Mathematical Behavior, 25(2), 136-152.

Previous research has shown an important link between a parent’s involvement with homework and a child’s academic performance. In addition to the time spent on the assignment, this study examined the quality of a mother’s mathematical content knowledge and her scaffolding during help sessions. The interactions of 158 mothers and their 5th grade children were recorded, following an individual briefing on the content with the parent. The adults also completed a questionnaire about topics such as their mathematics education and confidence in helping their child. Quantitative results showed that students spent an average of 23 minutes on mathematics homework daily with 4 minutes of help coming from each parent. Overall, there was a significant correlation between the mothers’ mathematics preparation and the quality of their content instruction and scaffolding. (B. Roper) 

Koehler, M., & Mishra, P. (2009). What is technological pedagogical content knowledge? Contemporary Issues in Technology and Teacher Education9(1), 60-70.

A well-known research for teacher knowledge is Shulman’s descriptions of PCK (Pedagogical Content Knowledge). The present paper suggests a framework for teacher knowledge for technology integration called technological pedagogical content knowledge(TPACK). The TPACK framework builds on Shulman’s description of PCK to describe how teachers’ understanding of educational technologies and PCK interact with one another to produce effective teaching with technology.  The authors describe why teaching with technology is difficult and what possibilities the TPACK framework can offer in teacher education.  (D. Shin)

Moyer, P. S., & Husman, J. (2006). Integrating coursework and field placements: The impact on preservice elementary mathematics teachers' connections to teaching. Teacher Education Quarterly, 33(1), 37-56.

The researchers in this study were interested in understanding how future educators are motivated, specifically in their own teacher preparation courses. The researchers hypothesized integrating methods classes with the school environment would make the pre-service teachers see themselves as future educators, thus motivating them to find value in their mathematics courses.  In this study there were two groups being observed. The first group took their methods courses at the college they attended, and only went to the school for the practicum experience.  Group two spent all day, from 7:30 to 3:00, at the school, taking their methods and practicum courses at the school. The researchers found that at the end of the course, the second group saw themselves more as “future professionals,” while the first group still felt like college students participating in classes. (S. Erwin)

Ryve, A., Nilsson, P. & Pettersson K. (October 2012). Analyzing Effective Communication in Mathematics Group Work: The Role of Visual Mediators and Technical Terms. Educational Studies in Mathematics, 82, 597-514. Retrieved from http://web.b.ebscohost.com/ehost/pdfviewer/pdfviewer?sid=cb114261-42cf-4a69-aa7b-d734791e02ae%40sessionmgr115&vid=1&hid=102 

In this research write up, the authors use their motivation for their findings to build off of previous studies about how communication can be effective in mathematics group work.  They focus on two main factors that seem as if they must coexist in order for communication to be effective: visual mediators and technical terms.  Visual mediators are symbolic artifacts in mathematics discourse such as algebraic expressions, tables, or graphs, and technical terms are the words that belong to a certain mathematical discourse.  The researchers studied two different groups in which they collected their data from.  One was a group of 12-13 year old sixth grade students working on a probability task involving a dice game and the other group was a small group of university students working on a proof by induction problem about a function.  (B. Koblitz)

van Es, E. A., & Sherin, M .G. (2009/2010). The influence of video clubs on teachers’ thinking and practice. Journal of Mathematics Teacher Education, 13, 155-176.  Retrieved from Springerlink. DOI: 10.1007/s10857-009-9130-3

I chose this article as part of my research about how video clubs are used in secondary mathematics. The researchers used qualitative data from teachers-- comments during meetings, self reports, and observation of instruction- to analyze the effects of the video club. Teachers shifted in their views of themselves as learners of mathematics through the interactions that took place in (and as a result of) the video club. There was a clear increase in focus on student thinking, although the authors note that more similar research needs to be conducted to validate this. (K. Patel)

Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities Research and Practice, 18(2), 121-131.

Due to the abstract nature of algebra, many students experience difficulty with this subject area in middle school classrooms. Using concrete models to supplement classroom lessons has been a difficult task to accomplish since some algebraic models are only suitable for modeling introductory concepts in algebra instead of the more advanced topics. In this study, the author compares traditional instruction in algebraic concepts to concrete-to-representational-to abstract (CRA) instruction. CRA instruction focuses on using concrete models to express algebraic ideas, which eventually lead to representing these ideas using pictures. After using these two representations of algebraic thought, students begin to reason abstractly about concepts learned in class. By using a pre-post-follow up design, the author was able document that the CRA instruction group outperformed the traditional instruction group during the post-test and follow up assessments. In addition to this, students in the CRA instruction group tended to make less errors when solving for variables as well. ​(K. Keels)

October 22 Entries

Jacobbe, T. (2012). Elementary School Teachers’ Understanding of the Mean and Median. International Journal of Science and Mathematics Education10(5), 1143-1161.

This article describes how elementary school teachers understand the mean and median. Three elementary school teachers participated in the study. The result is that they do not understand the material at sufficient depth to teach statistical content effectively. One of the optimistic things of this study is that all three teachers had acknowledged an awareness of their lack of content knowledge in the area of statistics at the end of the study, and they wished to receive professional development focused on this particular content strand. With the problematization they experienced, the teachers would be more likely to absorb the content introduced during sustained professional development. (D. Shin)

McGuire, P. & Kinzie, M. (2013). Analysis of place value instruction and development in pre-kindergarten mathematics. Early Childhood Education Journal, 41(5), 355-364.

This study is unique because most research on teaching place value in elementary school focuses on the upper grades, but in this article, the authors focus on pre-kindergarten students and teachers. The authors examine how the students learn two-digit place value, and they identify instances where the students struggled with the concepts that were taught. They also look for specific ways that the Pre-K teachers can support their students’ mathematical thinking and conceptual understanding in this area of mathematics. Instructional strategies used by high-quality teachers in a pre-kindergarten classroom setting are identified and discussed. (K. Keels)​

Plass, J. L., O’Keefe, P. A., Homer, B. D., Case, J., Hayward, E. O., Stein, M., & Perlin, K. (2013). The impact of individual, competitive, and collaborative mathematics game play on learning, performance, and motivation. Journal of Educational Psychology, 105(4), 1050-1066.

This study was conducted with middle school students in a northeastern city who were participating in an after school technology program. The researchers wanted to know if a computer game, FactorReactor, had varying effects on achievement goal orientation, situational interest, enjoyment, future intentions, and mathematics fluency, based upon how the students were allowed to play the game, whether as an individual, collaborative pair (2 students), or competitive (2 students).  The researchers found that students who played with someone else (whether competing or collaborating) had an increase in motivation to succeed than students who played alone, enjoyed the game more, and were more likely to recommend the game to a peer. The researchers struggled to determine whether the grouping affected students’ mathematical fluency. (S. Erwin)

Rittle-Johnson, B. & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91(1), 175-189.

This study investigates how the acquisition of conceptual knowledge can affect the problem-solving process of a mathematics student and vice versa. Prior research suggests that an understanding of concepts has a large impact on the development of related procedures. The researchers placed 48 fourth and fifth graders into three groups, based upon their results on a pretest over equivalent equations. One section received conceptual instruction, another procedural instruction, and there was a control group with no instruction. Results of the study showed that students in both of the instruction groups had an increase in conceptual knowledge. However, students in the conceptual group were able to generate multiple correct procedures rather than the only one that was taught in the other section. (B. Roper)

Schorr, R. Y., Firestone, W. A., & Monfils, L. (November 2003). State Testing and Mathematics Teaching in New Jersey: The Effects of a Test Without Other Supports. Journal for Research in Mathematics Education, 34(5), 373-405.

In this article, research findings are presented about if and how state tests can have an effect on the mathematics being taught.  The study takes place in New Jersey and is based mostly off of interview and observational data.  The state of New Jersey has a mathematics test that is said to be aligned with state and national standards called the Elementary School Performance Assessment (EPSA).  There are many factors that go into the effects of the test on teaching, and they can vary from state to state.  New Jersey specifically found that the teachers only felt slight pressure to bring up test scores because the state didn’t put as much emphasis on the consequences of test results as others may have.  This article concludes that there are still a lot of effects on teaching that have to be looked into that stem from mathematics testing.  However, four general changes were hinted out that teachers say they adjusted their teaching based on the test: more emphasis on student explanations of answers, using manipulatives, emphasizing problem solving, and having students write more about mathematics.  (B. Koblitz) 

Stockero, S.L., Peterson, B.E., Leatham, K.R., & Van Zoest, L.R. (2014). The "most" productive student mathematical thinking. The Mathematics Teacher, 108(4), 308-312. Retrieved from http://www.nctm.org/publications/article.aspx?id=43388

Incorporating student thinking into instruction can be challenging, but has a lot of potential for supporting student understanding. This article discusses recognizing student thinking and using it to build understanding. Not every piece of student thinking is practical or beneficial to build upon, so this article focuses on identifying those that are useful as Mathematical Opportunities in Student Thinking. To help teachers with this, a framework is discussed that provides three criteria that can be used to analyze a student's contribution -- student mathematical thinking, mathematically significant, pedagogical opportunity. (K. Patel)

Watt, H. M. G. (2005). Adolescent motivations for pursuing maths-related careers. Australian Journal of Educational and Developmental Psychology, 5, 107­116.

Watt (2005) uses the Expectancy-Value model developed by Eccles and colleagues to analyze interview data from 60 ninth grade students in Sydney, Australia. The goal was to explore why students plan to choose or not choose mathematics-related careers. Simultaneously, the author evaluates the exhaustiveness of the Expectancy-Value model. Students were grouped by mathematics performance and talent perceptions, with 20 students (10 boys and 10 girls) in each group. The results showed that all the responses fit the Expectancy-Value model. In addition, the most common reasons giving for choosing a mathematics-related career was the career happened to involve mathematics, not for the mathematics itself. The most common reason for not choosing mathematics-related career is that students had different interests, with lack of interest in mathematics as second most common. This study provides insight into student perceived influences on career-choices related to mathematics and confirms the exhaustiveness of the Expectancy-Value model. (J. Przybyla-Kuchek)



October 29 Entries

Artique, M. & Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDM, 45,(6), 797-810.    (Appendix)

Artique and Blomhøj layout the history of inquiry-based instruction in mathematics education though its commonalities in six well-established theories in mathematics education: problem-solving (PS), theory of didactical situations (TDS), realistic mathematics education (RME), modeling perspectives, anthropological theory of didactics (ATD), and dialogical and critical approaches. Its roots are traced back to Dewey’s philosophy of education and learning and inquiry-based education’s emergence in science education. As a supplement to this article, the authors have complied six inquiry-based activities, one representative of each of the theories mentioned above, and published them in an online appendix: http://link.springer.com/content/esm/art:10.1007/s11858-013-0506-6/file/MediaObjects/11858_2013_506_MOESM1_ESM.pdf (J. Przybyla-Kuchek)

Bray, W.S. (2011). A Collective Case Study of the Influence of Teachers' Beliefs and Knowledge on Error-Handling Practices During Class Discussion of Mathematics. Journal for Research in Mathematics Education, 42(1), 2-38.

My article last week focused on teacher response to student thinking in class. This article is related, with a focus on how teachers respond to errors made in class. Teachers' responses are tied closely to beliefs and mathematical knowledge. Teachers vary in how they intentionally use flawed solutions with the whole class, how errors promote conceptual understanding, and how their community affects errors being a part of class discussions. (K. Patel)

Ding, M. & Carlson, M. A. (2013). Elementary teachers’ learning to construct high-quality mathematics lesson plans: A use of the IES recommendations. The elementary school journal, 113(3), 359-385.

In this article, Ding and Carlson present a study that addresses the need for elementary school teachers to plan detailed lessons. The authors aim to use the Institute of Education Sciences (IES) recommendations in order to support teachers while they design lesson plans related to equivalence, inverse relationships, and arithmetic rules. During this study, three IES recommendations were emphasized during an intense summer course. 35 teachers participated in this study, and the progress made by those teachers was recorded using work samples and surveys. The results indicate that these teachers made significant progress towards crafting effective lesson plans. (K. Keels)

Gasco, J., & Villarroel, J. (2013). The motivation of secondary school students in mathematical word problem solving. Electronic Journal of Research in Educational Psychology, 12(1). 83-106.

The researchers were studying the effects of different types of problem solving methods in students when doing mathematical word problems. They wanted to determine if algebraic, arithmetic, mixed, or no strategy in solving mathematical word problems would have different degrees of motivation, value, cost, and self-efficacy in mathematics. The study concludes tha tstudents who solved the word problem algebraically were more motivated to learn the mathematics than their peers, and students who solved the word problem algebraically or mixed found more value in the mathematics being learned, and also had a higher score in self-efficacy ,thus resulting in higher motivation scores. (S. Erwin)  

Judson, E. (Fall 2007). Retaking a High Stakes Mathematics Test: Examination of School Interventions and Environments. American Secondary Education, 36(1), 15-30.

This study compares ten schools with much success on high stakes mathematics tests with ten schools with low success rates.  The purpose is to find out how high schools supported the students who were retaking the mathematics portion of a high stakes test.  If schools or students fail to meet the requirements of high stakes testing, the consequences can be detrimental and therefore, the interventions that take place to improve these test scores are important to research.  For this research, a survey was taken concerning the similarities and differences in interventions between schools.  For the most part, the differences found were at the school level and had to do with decisions made and shared responsibility.  It was also found that the top schools were able to dedicate more resources toward their interventions as well as had greater outside support.   (B. Koblitz) 

Makar, K. (2014). Young children's explorations of average through informal inferential reasoningEducational Studies in Mathematics86(1), 61-78.

This article described how young children can use informal inferential reasoning in an inquiry based learning environment to develop rich conceptions of average. This paper reported on a year 3 (age 8) class as they wrestle with the questions: “Is there a typicalheight for a person in year 3? If so, what is it?”The results showed that five key concepts of “typical” (average) emerged as students debated the inquiry question. They regarded “typical” as (1) a reasonable height, (2) the most common value or interval of data in the class, (3) the middle height, (4) the medium (normative) population height and (5) representative of a subpopulation.(D. Shin).

Yee, S. P. & Bostic, J. D. (2014). Developing a contextualization of students’ mathematical problem solving. Journal of Mathematical Behavior, 36, 1-19.

Rather than focusing on the problems themselves, this study investigates how students contextualize the processes they use to solve them. The factors of coherency and consistency were used as frameworks, as well as the Common Core Standards for Mathematical Practice. The interviews of three middle school and three high school students were analyzed where they were presented with challenging but appropriate mathematics problems. Researchers were looking for students’ use of metaphors and various mathematical representations. By examining the figurative language, the authors concluded that students often think of problems as containers of information. Another important result from the research was that students who employed non-symbolic strategies were more successful than those with symbolic approaches; however, the latter method was used more frequently. (B. Roper)



November 5 Entries

Cankoy, O. & Tut, M. A. (March-April 2005). High Stakes Testing and Mathematics Performance of Fourth Graders in North Cyprus. The Journal of Educational Research, 98(4), 234-243.

In this study, the authors examine how 4th grade students are affected by high stakes mathematical testing.  Not only the effect on the students is looked at, but also how the instruction driven by the standardized tests affected the students' mathematical understanding and performances.  Classes and teachers were observed to see how much class time was spent on test taking skills categories and were classified into three different groups based on the percent of class time spent on these skills.  A multiple choice Mathematical Performance Test was developed and given to the fourth grade North Cyprus students in the study.  Results of this study found that fourth graders better performed on routine mathematics problems than they did on nonroutine problems.  If instruction is test driven, this doesn't improve the students' higher order thinking skills in preparation for them to do well on the nonroutine story problems.  Based off of their findings, the authors give suggestions for better improving instruction in light of high stakes testing. (B. Koblitz)

Gürbüz, R., & Birgin, O. (2012). The effect of computer-assisted teaching on remedying misconceptions: The case of the subject “probability." Computers & Education58(3), 931-941.

This study investigates the effects of computer-assisted teaching on remedying misconceptions of probability concepts. Through previous studies, the authors dealt with three widespread misconceptions of probability: Probability Comparisons, Equiprobability, and Representativeness. They developed 12-items based on three misconceptions above and implemented them with 37 seventh-grade level students in a primary school in Turkey. By analyzing the results using pre- and post-test, they concluded that computer-assisted teaching was more effective than traditional methods to remedy misconceptions of probability and implied that the materials used in this study can also be applied in conjunction with several other instruction tasks. (D. Shin)

Kliman, M., Jaumot-Pascual, N., & Martin, V. (2013). How wide is a squid eye: Integrating mathematics into public library programs for the elementary grades. Afterschool Matters, 17. 9-15.

This is a report on a study conducted in eight libraries in low income areas. The researchers wanted to promote mathematics education and learning outside of the school environment; particularly in library programs. They developed a program that would be useful for educating children in a library, and offered it for free to the libraries. They found that after using this curriculum, librarians talked about mathematics more with students, integrated it more often into their lessons, and felt more confident in their mathematical ability. (S. Erwin)  

Lynch, K. & Star, J.R. (2014). Views of Struggling Students on Instruction Incorporating Multiple Strategies in Algebra 1: An Exploratory Study. Journal for Research in Mathematics Education, 45(1), 6-18.

Incorporating multiple strategies for solving is not a new idea in mathematics education. Many educators believe that this is only effective with students who excel in mathematics, not with those that are struggling in mathematics. This study interviews students after a year-long implementation of an algebra curriculum that looks at multiple methods for problems. The interviews with students identified as "struggling" led to positive support for multiple instructional strategies.   (K. Patel)

Schoenfeld, A. H. & Kilpatrick, J. (2013). A US perspective of inquiry-based learning in mathematics. ZDM, 45(6), 901–909.

Schoenfeld and Kilpatrick describe the past and current educational policies in the US and how the policies could affect an effort to implement Inquiry-Based Mathematics Education (IBME) on a large scale in the US, such as that done in Europe with the PRIMAS project. In particular, they describe the ability to implement such a program based on how well it would align with (1) perceived societal needs, (2) schooling traditions, (3) the specific framing of CCSS-M goals and (4) teacher professional development. Then, the authors describe six issues that one would face in trying to implement and IBME program on a large scale in the US in comparison to the successful implementation of PRIMAS.  (J. Przybyla-Kuchek)

Schukajlow, S. & Krug, A. (2014). Do multiple solutions matter? Prompting multiple solutions, interest, competence, and autonomy. Journal for Research in Mathematics Education, 45(4), 497-533.

There have already been several studies about the effects of multiple solution strategies on student achievement, but this article examines their impact on individual interest, competence, and autonomy. Students in six German classes were split into two groups each for five lessons; one with tasks that presented a clear solution method and the other with vague conditions. The students in the latter group showed a greater increase in their mathematics interest than those in the former group by the end of the study. They also developed more solution methods and felt more competent and autonomous in the subject than their counterparts. The tasks used in the study are provided in appendices. (B. Roper)   

Simon, M. A., Tzur, R., Heinz, K., & Kinzel, M. (2004). Explicating a mechanism for conceptual learning: Elaborating the construct of reflective abstraction. Journal of Research in Mathematics Education, 35(5), 305-329.

The authors of this article use Piagetian theory to develop a mechanism for mathematics conceptual learning that could inform how lessons should be designed for students. The mechanism presented by the researchers is highlighted and explained by using several student work samples. The lesson design proposed in this article illustrates the connection between Simon’s hypothetical learning trajectory and Piaget's reflected abstraction, assimilation, and accommodation. The investigators also outline the challenges of using Piaget’s work and the difficulties of addressing the learning paradox when attempting to explain how learning occurs. (K. Keels)



November 12 Entries

Andrá, C., Lindström, P., Arzarello, F., Holmqvist, K., Robutti, O., & Sabena, C. (2013). Reading mathematics representations: an eye-tracking studyInternational Journal of Science and Mathematics Education, 1-23.  (Published only on-line, not assigned to a regular print issue of the journal)

This article investigates whether there are differences between formulas and graphs in terms of the movement of eye and what entails such differences in terms of reading and understanding a mathematical text. Forty-six university students were participated in this study. Some tasks designed by researchers were provided with the students and students’ eye movements were recorded by using eye-tracking. The results of this study indicate that there is a meaningful difference between the ways one perceives formulas and graphs. These results can be analyzed by the cultural and social respective as well as by the differences between the nature of formulas and graphs (D. Shin).

Au, W. (2007).  High Stakes Testing and Curricular Control:  A qualitative metasynthesis. Educational Researcher, 36(5), 258-267.

The main focus of this research is answering the following question: "What, if any, is the effect of high stakes testing on curriculum?" The author uses qualitative metasynthesis from 49 previous studies in an attempt to answer this question using a complicated coding system. Au focuses on three components of curriculum: embodying content, knowledge form, and pedagogy. The results suggest a negative effect on curriculum with evidence of content being taught in pieces and teacher centered instruction in response to the high stakes test. However, there was also evidence that in some cases, classes were still student centered and content was integrated even with high stakes testing. Therefore, I don't think this research has really told us much that we don't already know. The main take away is that how high stakes tests are structured affects the curriculum in response to it, but again, this can be assumed without in depth research. (Brooke Koblitz)

Boaler, J., Wiliam, D, & Zevenbergen, R. (2000, March). The construction of identity in secondary mathematics education. Paper presented at the International Mathematics Education and Society Conference, Montechoro, Portugal.

The authors of this paper analyze interview data from 120 students in the UK and the US to argue the importance of identity for students learning of secondary school mathematics. The analysis revealed how students related to mathematics through their perceptions of mathematics and of themselves. Students’ talk about mathematics indicates that their interest in mathematics was rooted in how they see themselves and whether or not their perception of mathematics aligned with this sense of identity. Students who did not identify with mathematics tend not enjoy it; however, students who enjoyed mathematics did not express reasons involving their identity. This social perspective of learning would lead to changes of the definitions of success and failure in school mathematics. (J. Przybyla-Kuchek)

Lewis, K. E. (2014). Difference not deficit: Reconceptualizing mathematical learning disabilities. Journal for Research in Mathematics Education, 45(3), 351-396.

Two adult students who have been identified as having mathematical learning disabilities participate in tutoring sessions about fraction understanding. The 4 sessions did not help either student show growth from the pre-test to the post-test. The author uses the sessions to investigate how the students are conceptualizing fractions and the representations of fractions. This is related to the Vygotskian theory of viewing mathematical learning disabilities as differences and not as deficits. (K. Patel)

Norton, A. & Ambrosio, B. S. (2008). ZPC and ZPD: Zones of teaching and learning. Journal of Research in Mathematics Education, 39(3), 220-246.

Similarities and differences between the social construction of knowledge and radical constructivist theory are presented in this article, and Norton and Ambrosio go on to discuss two zones of learning proposed by these learning theories—the zone of proximal development (ZPD) and the zone of potential construction (ZPC). The authors examine whether operating within the two zones of learning has any effect on students’ ability to construct knowledge. The researchers explore the schemes available to two students while they reason about fraction concepts and attempt to construct schemes that allow them to conceptually understand equivalent fractions, non-unit fractions, improper fractions, and mixed numbers. (K. Keels)

Porter, M. K., & Masingila, J. O. (2000). Examining the effects of writing on conceptual and procedural knowledge in calculusEducational Studies In Mathematics42(2), 165-177.

This study investigated whether writing prompts in a collegiate calculus class can increase a student’s conceptual and procedural knowledge. The participants were students in two introductory calculus classes at a research university taught by the same professor; one section frequently used assignments with writing prompts while the other had standard activities. Both in-class and final examinations were analyzed for conceptual and procedural errors to use as data in the research. Results showed that there was no statistical difference in the number of errors committed by the two groups. The authors contradicted previous research by concluding that the benefit of writing in mathematics classrooms is communicating conceptual understanding rather than the act itself. (B. Roper)

Stewart, B. M., Cipolla, J. M., & Best, L. A. (2009). Extraneous information and graph comprehension: Implications for effective design choices. Campus-Wide Information Systems, 26(3), 191-200.

The researchers in this study were interested in understanding color and dimensionality’s effects on graphical understanding. They created a questionnaire with varying graphical representations, and tested for accuracy and reaction time. Overall, they found for the easiest questions, both 2D and 3D representations had similar percentages of accuracy. However, as the questions increased in difficulty, the accuracy of the answers for the 3D bar graphs diminished. In addition, they found the participants spent longer reading pie charts, but scored higher in accuracy than their comparative bar charts. They also found that color had less of an effect than they had anticipated. (S. Erwin)


November 19 Entries

Bishop, J. P. (2012). “She’s always been the smart one. I’ve always been the dumb one”: Identities in the mathematics classroom. Journal for Research in Mathematics Education, 43(1), 34–74.

Analyzing students’ mathematical identities can be difficult. In this article, Bishop creates a framework for analyzing student discourse in peer-to-peer interactions, through which students enact their mathematical identities. She then uses this framework to analyze the discourse between two seventh-grade girls during 13 days of mathematics classroom instruction involving group work. Bishop shows this framework to be one useful method of analyzing mathematical identity and how the results can be used to influence classroom instruction, group work, and mathematical education research on identity. (J. Przybyla-Kuchek)

Bright, G. W., Behr, M. J., Post, T. R., & Wachsmuth, I. (1988). Identifying Fractions on Number Lines. Journal for Research in Mathematics Education, 19(3), 215-232.

This study focused on small group and whole group instruction of fractions represented on a number line. Students had difficulties with this model and even more difficulties transferring understanding to new situations. Research found that students who had trouble partitioning and unpartitioning were not able to interpret and use number lines. Although the study is a bit dated, especially with more emphasis on number lines to represent fractions in the Common Core standards, it gives insight into student understanding of representations in early mathematics. These understandings have a lasting effect on understanding in secondary mathematics. (K. Patel)

Dempsey, J. V., Fisher III, S. H., & Hale, J. B. (1998). Quantitative graphical display use in asouthern U. S. school system. Research in the Schools, 5(1), 33-42

This is not an experiment, but a survey of how teachers perceive graph use in education. The researchers surveyed 429 teachers of elementary, middle, and high school and asked them to participate in a 78 item questionnaire about their perceptions of graphical displays in the school. They found many interesting responses, including the fact that only half the teachers had received training in using graphical displays, most teachers thought bar charts to be most important, and the amount of graphs used decreased as grade level increased. Overall this was a very informative, interesting report.
(S. Erwin)

Lin, J. J. H., & Lin, S. S. (2014). Cognitive load for configuration comprehension in computer-supported geometry problem solving: An eye movement perspective. International Journal of Science and Mathematics Education, 12(3), 605-627.

This study conducted two experiments. In particular, in the second experiment, the authors examined what cognitive load sources students explicitly noticed/used and whether eye movement patterns show sources of cognitive load in comprehending geometry configurations. Sixty three senior high school students in Taiwan participated in the second experiment in the study and five geometry problems were designed for them. The results from eye movement indicated that more attention and more time were given to reading the more difficult configurations than to reading the intermediate and easier configurations. In addition, the successful problem solvers did not fixate as long as the unsuccessful ones on the areas with crucial information (lengths of corresponding sides). The unsuccessful problem solvers scanned the whole diagram more extensively and attended to the intersection of overlapping, flip-over, upside-down and rotated triangles (D. Shin).

Madaus, G. F. (Spring 1998).The distortion of teaching and testing: High-stakes testing and instruction. Peabody Journal of Education, 65(3), 29-46.

In this article, the author discusses how high stakes tests can strongly affect how teachers teach and therefore, how the students learn.  He has a negative view on measurement driven instruction stemming from these tests.  He first discusses what a test is through definitions and speaking about validity.  Then, he gives 6 principles that he argues are impactful on measurement driven instruction.  Lastly, Madaus talks about what teachers can do to try to lessen the negative effects of high stakes tests, such as lowering the stakes.  This is more of a compilation of other literature and ideas about high stakes testing and instruction rather than a research experiment on its effects. (B. Koblitz)

Madaus, G., Russell, M., & Higgins, J.  (2009).   The paradoxes of high-states testing:  How they affect students, their parents, teachers, principals, schools, and society.   Charlotte, NC:   Information Age Publishing.

Olive, J. & Steffe, L. P. (2002). The construction of an iterative fractional scheme: The case of Joe. The Journal of Mathematical Behavior, 20(4), 413-437.

In this article, Olive and Steffe discuss how a child may reorganize number sequences in order to develop a Fractional Connected Number Sequence. This case study cites evidence supporting their claim using teaching episodes featuring a child named Joe. During the first year, the authors examine how Joe’s whole number operations compare to the partners that he works with, and during the second year, they concentrate on his treatment of fractions. In their study, Olive and Steffe note that Joe’s ability to act on composite units and use a splitting scheme when manipulating fractions were essential to his construction of a Fractional Connected Number Sequence. (K. Keels)

Yerushalmy, M. (2006). Slower algebra students meet faster tools: Solving algebra word problems with graphing software. Journal for Research in Mathematics Education, 37(5), 356-387.

This study investigates how and when lower-ability algebra students use graphing software while solving algebra problems in context. Yerushalmy analyzed interviews from three pairs of students that occurred intermittently from the 7th to 9th grade. These students were in the lower 25% of their grade based on exam scores and impressions by the interviewer. In the sessions, students completed application tasks about functions; they were provided with paper and access to graphing software. The data showed there were four primary processes related to the technology: using it to see a global picture, using it when the problem became difficult, working on paper due to computer limitations, and trying the problem on paper first. Another common trend in the results is students were initially hesitant to use the technology and were often encouraged by the interviewer to do so.   (B. Roper)