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Refereed Journal Article
Alvey, C., Hudson, R. A., Newton, J. & Males, L. M. (in press). Secondary pre-service teachers’ algebraic reasoning about linear equation solving. Issues in the Undergraduate Mathematics Preparation of School Teachers.
Publication year: in press
Abstract: This study analyzes the responses of 12 secondary pre-service teachers on two tasks focused on reasoning when solving linear equations. By documenting the choices PSTs made while engaging in these tasks, we gain insight into how new teachers work mathematically, reason algebraically, communicate their thinking, and make pedagogical decisions. We will share qualitative results from our examination of teacher knowledge through pre-service teachers’ explanations, models, language, and conjectures about student thinking.

Activities to support conceptual understanding of spatial measurement

Presentation
Males, L. M., Sweeny, S., Gilbertson, N., & Gonulates, F. (2011, April). Activities to support conceptual understanding of spatial measurement. Gallery session at the annual meeting of the National Council of Teachers of Mathematics, Indianapolis, IN.
Publication year: 2011

Exploring spatial measurement by attending to core conceptual principles

Presentation
Gonulates, F., Males, L. M., & Clark, D. (2011, April). Exploring spatial measurement by attending to core conceptual principles. Presentation at the annual meeting of the National Council of Teachers of Mathematics, Indianapolis, IN.
Publication year: 2011

Assessing curricular contributions to poor measurement learning

Presentation
Smith, J.P., Dietiker, L., Lee, K., Males, L. M., Figueras, H., & Mosier, M. (2008, April). Assessing curricular contributions to poor measurement learning. Research symposium at the research pre-session of the annual meeting of the National Council of Teachers of Mathematics, Salt Lake City, UT.
Publication year: 2008

Assessing curricular contributions to poor measurement learning

Presentation
Smith, J.P., Dietiker, L., Males, L. M., Figueras, H., Mosier, M., & Tan-Sisman, G. (2008, April). Assessing curricular contributions to poor measurement learning. Poster presentation at the annual meeting of the American Educational Research Association, New York, NY.
Publication year: 2008
Abstract: Twenty-five years of research has shown fundamental weaknesses in what U.S. students understand about spatial measurement without providing any compelling explanation for why this content proves so challenging for students. This paper reports the work of one project to assess how much weaknesses in the content of written mathematics curricula (K-8) contribute to the problem. Central to this work is a scheme for evaluating the capacity of representative elementary and middle school curricula to support students’ learning. Specifically, the purpose of this scheme is to support the identification of each instance where a curriculum provides students some opportunity to learn some bit of knowledge about spatial measurement. The paper will also relate this work to recent reconceptualizations of the conceptual-procedural knowledge distinction in mathematics education.

Adventures with mathematics

Presentation
Males, L. M., Cavazos, S., Frame, M., Gardner, R., Lantz, K., Rice, T., Scott, T., & Whaley, L. (2011, February). Adventures with mathematics. Presentation at the Math in Action conference of Grand Valley State University, Allendalle, MI.
Publication year: 2011

Exploring spatial measurement through a conceptual lens

Presentation
Males, L. M., & Gonulates, F., Sweeny, S., & Gilbertson, N. (2011, February). Exploring spatial measurement through a conceptual lens. Presentation at the Math in Action conference of Grand Valley State University, Allendalle, MI.
Publication year: 2011

Conceptual development of area

Presentation
Gonulates, F., Males, L. M., & Gilbertson, N. (2010, August). Conceptual development of area. Presentation at the annual meeting of the Michigan Council of Teachers of Mathematics, Macomb, MI.
Publication year: 2010

Getting down to the basics with length measurement

Presentation
Males, L. M., & Gonulates, F. (2009, August). Getting down to the basics with length measurement. Presentation at the annual meeting of the Michigan Council of Teachers of Mathematics, Macomb, MI.
Publication year: 2009

Strategies to help your measurement instruction measure up

Presentation
Smith, J.P., & Males, L. M. (2009, March). Strategies to help your measurement instruction measure up. Presentation at the Math in Action conference of Grand Valley State University, Allendale, MI.
Publication year: 2009

Measuring the measurement: An analysis of spatial measurement in elementary and middle school curricula within educational systems

Presentation
Smith, J.P., & Males, L. M. (2008, August). Measuring the measurement: An analysis of spatial measurement in elementary and middle school curricula within educational systems. Presentation at the annual meeting of the Michigan Council of Teachers of Mathematics, Holt, MI.
Publication year: 2008

How well do our texts introduce and define area?

Presentation
Smith, J.P., Dietiker, L., Lee, K., Figueras, H., & Males, L.M. (2007, August). How well do our texts introduce and define area? Presentation at the annual meeting of the Michigan Council of Teacher of Mathematics, Holt, MI.
Publication year: 2007

Activities to support conceptual understanding of spatial measurement

Presentation
Males, L. M., Sweeny, S., Gilbertson, N., & Gonulates, F. (2011, April). Activities to support conceptual understanding of spatial measurement. Gallery session at the annual meeting of the National Council of Teachers of Mathematics, Indianapolis, IN.
Publication year: 2011

Textual expression of knowledge in curricula: Illuminating opportunities to learn area measurement

Presentation
Males, L. M., & Gonulates, F. (2011, April). Textual expression of knowledge in curricula: Illuminating opportunities to learn area measurement. Poster presentation at the annual meeting of the Research Pre-session of the National Council of Teachers of Mathematics, Indianapolis, IN.
Publication year: 2011
Poster presentation

The STEM Project: Measuring capacity to learn measurement in curriculum

Presentation
Dietiker, L., & Males, L.M. (2008, February). The STEM Project: Measuring capacity to learn measurement in curriculum. Poster presentation at the annual research conference for the Center for the Study of Mathematics Curriculum, Phoenix, AZ.
Publication year: 2008
Poster presentation

Curricular Noticing: Theory on and Practice of Teachers’ Curricular Use

Refereed Book Chapter
Amador, J., Males, L. M., Earnest, D., & Dietiker, L. (in press).Curricular Noticing: Theory on and Practice of Teachers’ Curricular Use. Research in Mathematics Education book series (Springer)
Publication year: 2017-Expected
Abstract: In this study, we focus on one preservice teacher’s noticing of students’ mathematical and scientific thinking with an emphasis on how the acts of attending and interpreting can influence decisions about pedagogical actions. The study centers on an innovative field experience approach that incorporates lesson study in order to emphasize students’ thinking and its impact. Consequently, we were interested in understanding how teachers make decisions based on their noticing at three points in their career: preservice field experiences, student teaching, and their first year teaching. We used a case study approach to focus on one preservice teacher. Findings indicate that scaffolding PSTs to notice students’ mathematical and scientific thinking influenced how they notice and consider students’ thinking while teaching; however, the PSTs’ embedded noticing into their practice to different degrees. Results further indicate that supporting the development of noticing during field experiences has a positive impact on teachers when they are in their own classrooms. The study provides a unique contribution to the field as it incorporates both the mathematics and science teaching practices of the same six PSTs from their teacher education experience into their careers.

Using video of peer teaching to examine 6-12 preservice teachers’ noticing

Refereed Book Chapter
Males, L.M. (in press). Using video of peer teaching to examine 6-12 preservice teachers’ noticing. Research in Mathematics Education book series (Springer)
Publication year: 2017-Expected
Abstract: In this chapter, I describe what features secondary (7-12) mathematics preservice teachers (PSTs) identified as noteworthy in lessons taught by their peers in the context of a methods course. PSTs planned, taught, and reflected on at least two lessons from middle and high school reform-oriented materials taught to their peers across two semesters. Lessons were recorded and uploaded to VoiceThread, a web-based application that enables users to comment on video and these comments served as data for this study. Results indicated that across the two semesters, PSTs most frequently identified aspects related to communication, mathematics content, and classroom management with less evidence of attention to classroom environment and tasks. Although PSTs more often noted teacher talk or actions, rather than student talk or actions, the percentage of comments related to student talk or actions increased in the second semester. These results are significant in that they illustrate that PSTs can identify noteworthy features of classroom instruction and this assignment served as an opportunity for PSTs to do this.

Refereed Journal Article
Otten, S., Herbel-Eisenmann, B., & Males, L.M. (2010). Proof in algebra: An example of reasoning beyond examples. Mathematics Teacher, 103, 514-518.
Publication year: 2010
Abstract: A vignette from an early algebra class reveals a rich opportunity for generating proof before geometry.

Examining K-12 prospective teachers’ curricular noticing

Presentation
Males, L. M., Earnest, D., Dietiker, L., & Amador, J. (2015, November). Examining K-12 prospective teachers’ curricular noticing. Paper presented at the annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education, East Lansing, MI.
Publication year: 2015
This presentation explores the construct of curricular noticing, defined as the act of teachers making sense of the complexity of content and pedagogical opportunities in written or digital curricular materials (Dietiker, Amador, Earnest, Males, & Stohlmann, 2014), and reports the results of four exploratory studies aimed to examine the Curricular Noticing framework. Taken together, these studies capture work done with 62 PSTs in elementary and secondary mathematics methods courses at four universities. Findings illuminate what PSTs attend to in curriculum materials and how they interpret and respond to these materials. Irrespective of level (i.e., elementary, secondary) and materials, PSTs can learn to notice aspects of curriculum materials in order to make decisions about what to do and how to do it, and activities within methods courses can facilitate this development.

Examining 6-12 PSTs’ attention to curriculum materials when planning

Presentation
Males, L. M., Flores, M., & Williams, M. (2016, April). Examining 6-12 PSTs’ attention to curriculum materials when planning. Paper presentation at the annual Research Conference of the National Council of Teachers of Mathematics, San Francisco, CA.
Publication year: 2016
Abstract: This session describes an exploratory study that examines one aspect of Curricular Noticing. Specifically, what did secondary PSTs attended to in their curriculum materials when planning a lesson from a reform-oriented curriculum?

Examining K-12 prospective teachers’ curricular noticing

Refereed Proceedings Article
Males, L. M., Earnest, D., Dietiker, L., & Amador, J. (2015). Examining K-12 prospective teachers’ curricular noticing. In T.G. Bartell, K. N. Bieda, R. T. Putman, K. Bradfield, & H. Dominguez (Eds.), Proceedings of the 37th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. East Lansing, MI: Michigan State University.
Publication year: 2015
Abstract This paper explores the construct of curricular noticing, defined as the act of teachers making sense of the complexity of content and pedagogical opportunities in written or digital curricular materials (Dietiker, Amador, Earnest, Males, & Stohlmann, 2014), and reports the results of four exploratory studies aimed to examine the Curricular Noticing framework. Taken together, these studies capture work done with 62 PSTs in elementary and secondary mathematics methods courses at four universities. Findings illuminate what PSTs attend to in curriculum materials and how they interpret and respond to these materials. Irrespective of level (i.e., elementary, secondary) and materials, PSTs can learn to notice aspects of curriculum materials in order to make decisions about what to do and how to do it, and activities within methods courses can facilitate this development.

The treatment of area measurement in U.S. elementary mathematics curricula: Are students’ opportunities to learn sufficient?

Refereed Journal Article
Smith, III, J. P., Males, L. M., & Gonulates, F. (in press). The treatment of area measurement in U.S. elementary mathematics curricula: Are students’ opportunities to learn sufficient? Mathematical Thinking and Learning
Impact Factor: 1.333 (© 2016 Thomson Reuters, Journal Citation Reports®)
Publication year: 2016
Abstract Research has found that elementary students face five main challenges in learning area measurement: (a) Conserving area as a quantity, (b) understanding area units, (c) structuring rectangular space into composite units, (d) understanding area formulas, and (e) distinguishing area and perimeter. How well do elementary mathematics curricula address these challenges? A detailed analysis of three U.S. elementary textbook series revealed systematic deficits. Each presented area measurement in strongly procedural terms using a shared sequence of procedures across grades. Key conceptual principles were infrequently expressed and often well after related procedures were introduced. Particularly weak support was given for understanding how the multiplication of lengths produces area measures. The results suggest that the content of written curricula contributes to students' weak learning of area measurement.

The introduction of proof in secondary geometry textbooks

Refereed Journal Article
Otten, S., Males, L. M., & Gilbertson, N. J. (2014). The introduction of proof in secondary geometry textbooks. International Journal of Educational Research, 64, 107–118.
Publication year: 2014
Abstract Explicit reasoning-and-proving opportunities in the United States are often relegated to a single secondary geometry course. This study analyzed the reasoning-and-proving opportunities in six U.S. geometry textbooks, giving particular attention to the chapter that introduced proof. Analysis focused on the types of reasoning-and-proving activities expected of students and the type of mathematical statement around which the reasoning- and-proving took place, be it general or particular. Results include the fact that reasoning-and-proving opportunities in student exercises were predominantly of the particular type, whereas textbook exposition most commonly had general statements. Within the chapters introducing proof, opportunities for students to develop proofs were less common than exercises involving conjectures and statements or exercises about the reasoning-and-proving process. Opportunities to reflect on the reasoning-and-proving process were prevalent in the introduction chapters, though rare in the remainder of the books.

Planning with curriculum materials: An analysis of teachers’ attending, interpreting, and responding.

Refereed Proceedings Article
Males, L. M., Flores, M. Ivins, A., Smith, W. M., Lai, Y., & Swidler, S. (in press). Planning with curriculum materials: An analysis of teachers’ attending, interpreting, and responding. Proceedings of the 38th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Tucson, AZ.
Publication year: 2016

Abstract

Curriculum materials are integral to the teaching and learning of mathematics and have been described as having the most influence on what teachers actually plan for and enact in their classrooms. In this paper we describe how two teachers with varying years of experience, but similar experiences with curriculum materials, use curriculum materials in planning. Through a semi-structured think aloud interview, we describe what teachers attended to, how they interpreted what they attended to, and how they decided to respond to the curriculum materials when planning a hypothetical lesson on slope.

The mathematical nature of reasoning-and-proving opportunities in geometry textbooks

Refereed Journal Article
Otten, S., Gilbertson, N. J., Males, L. M., & Clark, D. L. (2014). The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. Mathematical Thinking and Learning, 16, 51–79. doi:10.1080/10986065.2014.857802
Impact Factor: 1.333 (© 2016 Thomson Reuters, Journal Citation Reports®)
Publication year: 2014

Abstract

International calls have been made for reasoning-and-proving to permeate school mathematics. It is important that efforts to heed this call are grounded in an understanding of the opportunities to reason-and-prove that already exist, especially in secondary-level geometry where reasoning-and-proving opportunities are prevalent but not thoroughly studied. This analysis of six secondary-level geometry textbooks, like studies of other textbooks, characterizes the justifications given in the exposition and the reasoning-and-proving activities expected of students in the exercises. Furthermore, this study considers whether the mathematical statements included in the reasoning-and-proving opportunities are general or particular in nature. Findings include the fact that the majority of expository mathematical statements were general, whereas reasoning-and-proving exercises tended to involve particular mathematical statements. Although reasoning-and-proving opportunities were relatively numerous, it remained rare for the reasoning-and-proving process itself to be an explicit object of reflection. Relationships between these findings and the necessity principle of pedagogy are discussed.

 Award: Named most read article in Mathematical Thinking and Learning in 2014. Currently the 10th most read article in Mathematical Thinking and Learning

Curricular Treatments of Length Measurement in the United States: Do They Address Known Learning Challenges?

Refereed Journal Article
Smith III, J. P., Males, L. M., Dietiker, L. C., Lee, K. & Mosier, A. (2013). Curricular Treatments of Length Measurement in the United States: Do They Address Known Learning Challenges? Cognition and Instruction, 31, 388-433, DOI: 10.1080/07370008.2013.828728. http://dx.doi.org/10.1016/j.ijer.2013.08.0066
Impact Factor: 2.172 (© 2016 Thomson Reuters, Journal Citation Reports®)
Publication year: 2013
Extensive research has shown that elementary students struggle to learn the basic principles of length measurement. However, where patterns of errors have been documented, the origins of students’ difficulties have not been identified. This study investigated the hypothesis that written elementary mathematics curricula contribute to the problem of learning length measurement. We analyzed all instances of length measurement in three mathematics curricula (grades K–3) and found a shared focus on procedures. Attention to conceptual principles was limited overall and particularly for central ideas; conceptual principles were often presented after students were asked to use procedures that depended on them; and students often did not have direct access to conceptual principles. We also report five groupings of procedures that appeared sequentially in all three curricula, the conceptual principles that underlie those procedures, and the conventional knowledge that receives substantial attention by grade 3.

Connecting Research to Practice: Reasoning-and proving opportunities in geometry textbooks

Refereed Journal Article
Gilbertson, N. J., Otten, S., Males, L. M., & Clark, D. L. (2013). Connecting Research to Practice: Reasoning-and proving opportunities in geometry textbooks. Mathematics Teacher, 107, 138-142.
Publication year: 2013

Educative supports for teachers in middle school mathematics curriculum materials: What is offered and how is it expressed?

Refereed Proceedings Article
Males, L.M. (2012). Educative supports for teachers in middle school mathematics curriculum materials: What is offered and how is it expressed? In L. R.Van Zoest, J. Lo, & J. L. Kratky (Eds.), Proceedings of the 34th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 96-99). Kalamazoo, MI: Western Michigan University.
Publication year: 2012
Abstract: In this paper I describe opportunities for teacher learning present in four middle school curricular series in the areas of introduction to variable and geometric transformations. I focus on one part of my analysis, the description of the opportunities present for developing Subject Matter Knowledge, Pedagogical Content Knowledge (for Topics and Practices), and Curricular Knowledge. My results indicated that opportunities for teachers’ development of Pedagogical Content Knowledge for Practices or Curricular Knowledge were most prevalent, whereas Subject Matter Knowledge was the least prevalent. In particular, opportunities lacked rationale guidance, or guidance that enables teachers to develop an understanding of why particular mathematical or pedagogical approaches might be appropriate.

Textual expression of area measurement in elementary curricula: Illuminating opportunities to learn

Refereed Proceedings Article
Gonulates, F., & Males, L.M. (2011). Textual expression of area measurement in elementary curricula: Illuminating opportunities to learn. In Ubuz, B. (Ed.). Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 441-448). Ankara, Turkey: PME.
Publication year: 2011
Abstract: This study explores opportunities to learn area measurement content through an examination of the textual elements in three widely used U.S. elementary curriculum materials. We focus on what the textual elements illuminated in regards to students’ access to area measurement concepts and procedures by describing the knowledge expressed and the curricular voice. We then provide a description of the textual expression of elements that are critical to building an understanding of area measurement. By examining the textual elements, we were able to describe the opportunity to learn area measurement. Our analysis indicated that while students were spoken to, they were often asked to “do” rather than “know.” While there were instances of underlying concepts, students did not often have direct access to them.

Reasoning-and-proving in geometry textbooks: What is being proved?

Refereed Proceedings Article
Otten, S., Gilbertson, N.J., Males, L.M., & Clark, D.L. (2011). Reasoning-and-proving in geometry textbooks: What is being proved?. In L. R. Wiest, & T. Lamberg (Eds.), Proceedings of the 33rd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 347-355). Reno, NV: University of Nevada, Reno.
Publication year: 2011
Abstarct: As calls are made for reasoning-and-proving to permeate school mathematics, several textbook analyses have been conducted to identify reasoning-and-proving opportunities outside of high school geometry. This study looked within geometry, examining six geometry textbooks and characterizing not only the justifications given and the reasoning-and-proving activities expected of students but also the nature of the mathematical statements around which reasoning-andproving takes place. The majority of reasoning-and-proving exercises focused on particular mathematical statements, whereas the majority of expository mathematical statements were general in nature. Although reasoning-and-proving opportunities were numerous, it remained rare for reasoning-and-proving to be made an explicit object of reflection.

Exploring U.S. textbooks’ treatment of the estimation of linear measurement

Refereed Journal Article
Chang, K., Males, L.M., Mosier, A., & Gonulates, F. (2011). Exploring U.S. textbooks’ treatment of the estimation of linear measurement. ZDM-The International Journal on Mathematics Education. doi:10.1007/s11858-011-0361-2.
Impact Factor: 1.96 (© 2016 Thomson Reuters, Journal Citation Reports®)
Publication year: 2011
Abstract: Learning to estimate a linear measurement is critical in becoming a successful measurer. Research indicates that the teaching of the estimation of linear measurement is quite open and that instruction does not make explicit to students how to carry out estimation work. Because written curriculum has been identified as one of the main sources affecting teachers’ instruction and students’ learning, this study examined how estimation of linear measurement tasks were presented to students in three US elementary mathematics curricula to see how much and in what ways these tasks were presented in an open manner. The principal result was that the length estimation tasks were frequently not explicit about which attribute of the object to measure and the requested level of precision of the estimate. Length estimation tasks were also left more open than other measurement tasks like measuring length with rulers.

Concentrating on nets

Other Publication
Males, J. R., & Males, L.M., (2010). Concentrating on nets. In L. M. Males & J. R. Males (Eds.), Adventures with mathematics: Climbing from grade 6 to grade 7 (pp. 16-20). Ann Arbor, MI: Michigan Council of Teachers of Mathematics.
Publication year: 2010

Adventures with mathematics: Climbing from grade 6 to grade 7

Other Publication
Males, L.M., Males, J.R., & Beckmann, C. (Eds.). (2010). Adventures with mathematics: Climbing from grade 6 to grade 7. Ann Arbor, MI: Michigan Council of Teachers of Mathematics.
Publication year: 2010

Challenges of critical colleagueship: Examining and reflecting on study group interactions

Refereed Journal Article
Males, L. M., Otten, S., & Herbel-Eisenmann, B. (2010). Challenges of critical colleagueship: Examining and reflecting on study group interactions, Journal of Mathematics Teacher Education, 13, 459-471. Special Issue on Teacher Change. DOI 10.1007/s10857-010-9156-6.
Impact Factor: 1.259 (2014 Source Normalized Impact per Paper)
Publication year: 2010
Abstract: This article examines mathematics teacher collegiality by focusing on both the ways in which teachers interacted as critical colleagues in a long-term professional development project and the evolving role of the teacher–educator–researcher as the facilitator of this project. The professional development collaboration comprised two phases: one focused on reading classroom discourse literature and one focused on supporting each other through cycles of action research related to mathematics classroom discourse. Lord’s (1994) critical colleagueship framework is used to examine how a study group of middle-grades (ages 11–16) mathematics teacher–researchers took (or did not take) a more critical stance toward their own teaching practice and that of their colleagues. We found that challenging interactions were related to instances in which the teachers interacted as critical colleagues and were marked by particular features including the use of particular words and the use of personal experience as a form of evidence. We present the ways in which we came to understand what it might look like to scrutinize one’s practice and findings related to the development of this type of collegiality across the two different phases of this project. We end with a section in which the teacher–educator–researcher who facilitated the professional development project reflects on the ways in which the analysis caused her to reconsider both the nature of argumentation in mathematics study group settings and what implications this has with respect to her own practice as a facilitator.

Dictionary of Mathematics Terms [Review of the book Barron’s Dictionary of Mathematics Terms, by Douglas Downing]

Refereed Book Review
Males, L.M. (2010). Dictionary of Mathematics Terms [Review of the book Barron’s Dictionary of Mathematics Terms, by Douglas Downing]. Mathematics Teaching in the Middle school, 16, 61.
Publication year: 2010

An argument for taking up similar work

Refereed Book Chapter
Herbel-Eisenmann, B., Cirillo, M., & Males, L.M. (2009) An argument for taking up similar work. In B. Herbel-Eisenmann & M. Cirillo (Eds.), Promoting purposeful discourse: Teacher research in mathematics classrooms (pp. 219-232). Reston, VA: National Council of Teachers of Mathematics.
Publication year: 2009

Confronting practice: Critical colleagueship in a mathematics teacher study group

Refereed Proceedings Article
Males, L. M. (2009). Confronting practice: Critical colleagueship in a mathematics teacher study group. In S. L. Swars, D. W. Stinson, & S. Lemons-Smith (Eds.), Proceedings of the 31st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 929-937). Atlanta, GA: Georgia State University.
Publication year: 2009
Abstract: This study investigates the development of “critical colleagueship” (Lord, 1994) by eight middle-grades mathematics teachers participating in a teacher study group as part of a project focused on improving mathematics classroom discourse. Analysis of the action research phase of this project indicated that aspects of critical colleagueship, such as self-reflection, openness to new ideas, the capacity for empathetic understanding, and the ability to reject flimsy reasoning were exhibited by the teachers. These aspects were manifested in three interaction patterns ranging from the common patterns of praise and advice-giving to the uncommon pattern of teachers engaging as challenging colleagues.