**EJC: PUBLICATIONS**

~Eric Bach

**•Articles in Refereed Journals (30)**

**•Articles in Non-Refereed Journals, Magazines and Periodicals (22)**

**•Encyclopedia Entries (3)**

**•Articles in Refereed Conference Proceedings (25)**

**•Articles in Non-Refereed Conference Proceedings (13)**

**•Books and Monographs (7)**

**•Book and Monograph Matter (22)**

**•Chapters in Edited Books (5)**

**•Edited Volumes (13)**

**•Editorials (14)**

**•Dissertation (1)**

**•eBooks (3)**

**•Book Reviews (1)**

**•Other Publications (5)**

**Articles in Refereed Journals (30)**

30. Chernoff, E. J. (In Press). The Canadian Math Wars. Canadian Journal of Science, Mathematics and Technology Education [Special Issue: Mathematics Education in the News], 16(1), xxx-xxx. DOI: 10.1007/s42330-018-0037-9

29. Chernoff, E. J. (2018). If Trump were an applicant to your mathematics education program, would you accept him? A response to Rodriguez, Kitchen and Harding. for the learning of mathematics: an international journal of mathematics education, 38(2), 27.

28. Chernoff, E. J., Russell, G. L., Vashchyshyn, I., Neufeld, H., Banting, N. (2017). There is no evidence for order mattering; therefore, order does not matter: An appeal to ignorance. Avances de Investigación en Educación Matemática, 11, 5-24.

Abstract. *Within the limited field of research on teachers’ probabilistic knowledge, incorrect, inconsistent and even inexplicable responses to probabilistic tasks are most often accounted for by utilizing theories, frameworks and models, which are based upon heuristic and informal reasoning. More recently, the emergence of new research based upon logical fallacies has been proving effective in explaining certain normatively incorrect responses to probabilistic tasks. This article contributes to this emerging area of research by demonstrating how a particular logical fallacy, known as “an appeal to ignorance,” can be used to account for a specific set of normatively incorrect responses provided by prospective elementary and secondary mathematics teachers to a new probabilistic task. It is further suggested that a focus on the classical approach to teaching theoretical probability contributes to the use of this particular logical fallacy.*

27. Chernoff, E. J. (2017). Solving Equations: A Make-Work Project for Math Teachers and Students. Journal of Humanistic Mathematics, 7(1), 251-262. doi: 10.5642/jhummath.201701.19

Abstract.* The purpose of this article is to share a particular view I have towards solving equations in the school mathematics classroom. Essentially, I contend that solving equations in the math classroom is a make-work project for math teachers and students. For example, math teachers take a predetermined value, which makes a statement true, and then make it harder and harder and harder for their students to determine the value that makes the statement true. However, they do so with the explicit purpose of teaching (then having) their students reveal the solution that they themselves have concealed. Stated in make-work project parlance, the math teacher digs a hole with the explicit purpose of teaching and the having students fill the hole they dug.*

26. Chernoff, E. J. (2017). Numberlines: Hockey Line Nicknames Based on Jersey Numbers. The Mathematics Enthusiast, 14(1-3), 371-386.

Abstract. *The purpose of this article, in general, is to expound Chernoff’s (2016) notion of numberlines, that is, hockey line nicknames based on jersey numbers. The article begins with a brief discussion of the history of hockey line nicknames, which allows for the parsing of numberlines and quasi-numberlines (nicknames based on numbers associated with hockey players). Focusing, next, on jersey number restrictions for the National Hockey League (NHL), a repeated calculation of the number of possible numberlines winnows down the number from a theoretical upper bound to a practical upper bound. Moving beyond the numbers, the names of natural numbers – those with a certain panache (e.g., Untouchable, McNugget, Frugal, Hoax, Narcissistic, Unhappy, Superperfect and Powerul numbers) – act as a gateway to the notion of numberlining, the process of attempting to coin a numberline. Two particular examples, The Powers Line and The Evil Triplets provide a window into the process of numberlining. Prior to concluding remarks, which explain how numberlines and numberlining fall in line with the NHL’s recent embrace of fans’ use of social media, the article details how adopting hockey line nicknames based on jersey numbers can be used as a possible venue to rename questionable hockey line nicknames. *

25. Neufeld, H. L., Vashchyshyn, I. I., & Chernoff, E. J. (2016, Autumn). Building Bridges: Barriers to Parent Engagement faced by Secondary Mathematics Teachers [Themed Issue: Linking Education and Community: Present and Future Possibilities]. LEARNing Landscapes, 10(1), 199-214.

**Abstract.** *Although parent engagement is widely supported by research, it is largely absent in the secondary mathematics classroom. Limited preservice teacher education and perceptions surrounding teacher professionalism are discussed as barriers to engaging parents. Math teachers are additionally inhibited by the antagonistic portrayal of parents in the literature and in the media, effectively alienating parents in the minds of teachers. We suggest a shift in the language used to discuss math education and the positioning of parents regarding knowledge as a way to enable parent engagement and build relationships of trust, which can transform otherwise difficult exchanges between teachers and parents.*

24. Vashchyshyn, I., Neufeld, H. & Chernoff, E. J. (2016). A case for humility in the mathematics classroom. Ontario Mathematics Gazette, 54(4), 11-15.

Abstract.* Humility is not a virtue frequently associated with good teaching, and much less with mathematics – a subject considered by many, if not most, to be a serious and strict science that offers little room for ‘soft’ values like humility. Indeed, as Chancellor and Lyubomirsky (2013) write, humility may be the most overlooked and underappreciated virtue of all in any context. Perhaps this is due to the fact that humility is often conflated with low self-esteem, even in the psychology literature (see, e.g., Weiss & Knight, 1980), or that humility feels like a square peg in a round hole within a culture that shies away from the notion of fallibility, preaching boundless self-assurance as a means to achieve success. However, as countercultural as this virtue may be, it need not suggest weakness or docility: On the contrary, true humility requires emotional resilience and a secure sense of self. And as frightening as the prospect of admitting that one is “out of moves” may be, humility on the teacher’s part may have a critical role to play in enriching students’ learning and enjoyment of mathematics, as well as in understanding the nature of mathematical activity itself. In this essay, we illustrate the point through two vignettes.*

23. Vashchyshyn, I. & Chernoff, E. J. (2016). A formula for success? An examination of factors contributing to Quebec students’ high achievement in mathematics. Canadian Journal of Education/Revue canadienne de l'éducation, 39(1), 1-26.

Abstract.* As the only province having achieved above the Canadian average in the latest PISA assessment and with an average score that was surpassed by only five other participating countries, Quebec has recently taken center stage as Canada’s superstar in the teaching and learning of mathematics. However, there has been relatively little discussion surrounding why Quebec students have been consistently successful in their mathematical endeavors. In this essay, the authors examine several possible influences, including an emphasis on problem solving, recreational mathematics activities, intensive teacher education programs, and active mathematics teacher associations. Our aim is to begin a conversation surrounding the following question: what can we, as mathematics teachers, learn from our neighbours in la belle province?*

22. Chernoff, E. J., Mamolo, A. & Zazkis, R. (2016). An investigation of the representativeness heuristic: the case of a multiple choice exam. EURASIA Journal of Mathematics, Science and Technology Education, 12(1), 1—23. doi: 10.12973/eurasia.2016.1252a

Abstract.* By focusing on a particular alteration of the comparative likelihood task, this study contributes to research on teachers’ understanding of probability. Our novel task presented prospective teachers with multinomial, contextualized sequences and asked them to identify which was least likely. Results demonstrate that determinants of representativeness (featured in research on binomial, platonic sequences) are present in the current situation as well. In identifying a variety of context-related features influencing teachers’ choices, we suggest the context in which tasks are presented significantly influences probabilistic judgments; however, contextual consideration also provides researchers with potential difficulties for analyzing results. In addition, we identify strands for further research of contextual influence.*

21. Russell, G. L. & Chernoff, E. J. (2016). The Transreform Approach to the Teaching and Learning of Mathematics: Re-viewing the Math Wars. Far East Journal of Mathematical Education, 16(1), 69-109. doi: 10.17654/ME016010069

Abstract.* It started with the question “How can (and will) teachers of mathematics in Canada both change the strategies and approaches that they use to teach mathematics AND infuse First Nations, Métis and Inuit content, perspectives, and ways of knowing into that teaching?” and with the then unjustified belief that the two undertakings were somehow connected. This study uses the theoretical lenses of two different worldviews to analyze the new teaching and learning expectations proposed within the Western and Northern Canadian Protocol Common Curriculum Framework documents to explore the possibility of such connections. Further, the results of our analysis give rise to a new approach to the teaching and learning of mathematics that resides beyond reform approach on the math wars continuum: the transreform approach to the teaching and learning of mathematics.*

20. Bond, G. & Chernoff, E. J. (2015). Mathematics and Social: A Symbiotic Pedagogy. Journal of Urban Mathematics Education (JUME), 8(1), 24-30.

Abstract.* Mathematics can be defined as “the science of pattern and order”. But because there is often a perceived spectrum of approachability to mathematics (based on common mis- conceptions that envision the subject as a sort of elitist wizardry) it is important to bear in mind different definitions of mathematics when exploring applications of mathematics in the classroom. This is especially true when considering the instruc- tion of mathematics for social justice.*

19. Chernoff, E. J. & Chernoff, J. W. (2015). Revealing subjective probability in the middle and high school mathematics classroom. Ontario Mathematics Gazette, 53(4), 30-35.

Abstract.* Individuals, those well versed in probability and statistics, understand that there are different interpretations of probability. Of the big three, that is, classical, frequentist and subjective probability, middle and high school mathematics students are given ample opportunities to explore classical and frequentist probability and, further, connections between the two interpretations. However, for certain reasons (detailed in this article) the same cannot be said for the subjective interpretation of probability. As a result, the purpose of this article is to share an innovative approach to the teaching and learning of a central tenet of subjective probability in the middle and high school math class.*

18. Chernoff, E. J. & Mamolo, A. (2015) Unasked but answered: comparing the relative probabilities of coin flip sequence attributes. Canadian Journal of Science, Mathematics and Technology Education, 15(2), 186-202. doi: 10.1080/14926156.2015.1031410

Abstract.* The objective of this article is to contribute to research on teachers’ probabilistic knowledge and reasoning. To meet this objective, prospective mathematics teachers were presented coin flip sequences and were asked to determine and explain which of the sequences was least likely to occur. This research suggests that certain individuals, when presented with a particular question, answer different questions instead. More specifically, we found that participants, instead of making the intended relative probability comparison, compared the relative probability of a number of particular attributes associated with coin flip sequences. Further, we interpret participants’ attempts to reduce levels of abstraction in order to reason about probability, in a relative sense. Embracing the research literature suggesting that responses reflect individuals’ understanding of the question they were asked, this article suggests potential questions that participants have not been asked, but are answering. In doing so, this article will suggests that participants are providing reasonable relative probability comparisons for questions that are unasked. Finally, implications for future research are also discussed.*

17. Higgs, N. & Chernoff, E. J. (2014). Content knowledge for teaching mathematics: How much is needed and are (Saskatchewan) teacher candidates getting enough? delta-K: Journal of the Mathematics Council of the Alberta Teachers' Association, 52(1), 17-21.

Abstract.* What makes a good math teacher? A common dichotomy is often brought up when discussing this particular question – one that pits two hypothetical math teachers against each other. Is the teacher who is an expert at math but not very skilled in pedagogy better than the teacher who knows very little about math but is very skilled in pedagogy? Different views and philosophies will be thrown around and argued, usually with both sides eventually conceding that you need at least a decent understanding in both math and pedagogy in order to be an effective math teacher. Yet the debate over how much math knowledge teachers and teacher candidates should have to effectively teach math continues. This article aims to better understand how to answer this particular question by analyzing the research on the topic of how much mathematical concept knowledge a teacher candidate should have. We begin with a review of the research and theory on the importance of mathematical knowledge for teacher candidates, then analyze how the research and theory fits in with current education that teacher candidates are receiving (with a special focus on the University of Saskatchewan and local School Divisions), and will conclude with a discussion of the implications of this analysis for aspiring math teachers.*

16. Brandt, A. & Chernoff, E. J. (2014). The Importance of Ethnomathematics in the Math Class. The Ohio Journal of School Mathematics [Journal of the Ohio Council of Teachers of Mathematics], 71(Fall), 31-36.

Abstract.* We contend that the teaching and learning of mathematics should reflect and embrace the cultural diversity found in our mathematics classrooms, and in our increasingly interconnected world. The goal of this article is to convey a simple message: ethnomathematics, that is, culturally based mathematics, should be (further) integrated into the mathematics classroom. To achieve this goal we discuss what ethnomathematics is and why it should be (further) incorporated into mathematics curricula. We also present examples of ethnomathematics in the math class, some of the arguments against inclusion of ethnomathematics into the curricula, as well as some ways in which these arguments can be successfully countered. Ultimately, we hope to demonstrate that ethnomathematics, which has the potential to show our students multicultural views of mathematics, may help students develop a greater interest in mathematics.*

15. Stone, M. & Chernoff, E. J. (2014). An examination of math anxiety research. Ontario Mathematics Gazette, 52(4), 29-31.

Abstract.* Math anxiety can be defined as a feeling of nervousness, unease, or tension that “interferes with math performance” (Ashcraft, 2002, p. 181). Taking liberties with this definition, math anxiety can be considered within the category of transmissible or communicable “diseases,” which may lie dormant within individuals for many years. From this new perspective, this article will provide a brief look at the characteristics of this illness, will outline some of the most common symptoms exhibited by hosts, and detail some of the recent advances in science that aim to manage or control and alleviate some of these damaging symptoms. Alternatively stated, this article is an “examination” of math anxiety research. Further, we provide a brief outline of possible measures that can be taken to prevent the further spread of this infectious disease.*

14. Chernoff, E. J. (2014). What would David Wheeler Tweet? *For the Learning of Mathematics**, 34*(1), 8.

Abstract. *What would David Wheeler Tweet?*

13. Chernoff, E. J. (2013). Probabilistic relativism: a multivalentological investigation of normatively incorrect relative likelihood comparisons [Special issue: Postmodern Mathematics/Mathematics Education]. Philosophy of Mathematics Education Journal, 27, 1-30. Retrieved from http://people.exeter.ac.uk/PErnest/pome27/index.html

Abstract.* This research continues the longstanding tradition of investigating relative likelihood comparisons. Respondents are presented with sequences of heads and tails derived from flipping a fair coin five times, and asked to consider their chances of occurrence. An iteration of the task, which maintains the ratio of heads to tails in all of the sequences presented, provides unique insight into individuals’ normatively incorrect relative likelihood comparisons. In order to reveal the aforementioned insight, this research, based upon participants’ response justifications, presents unconventional partitions of the sample space, which are organized according to switches, longest run and switches and longest run. In doing so, it will be shown that normatively incorrect responses to the task are not necessarily devoid of correct probabilistic reasoning. To accurately render the data gathered from 239 prospective mathematics teachers, an original theoretical framework (the meta-sample-space) will be used with a new method (event-description-alignment) to demonstrate, that is model, that certain individuals base their comparisons of relative likelihood according to a subjective organization of the sample space, that is, a subjective-sample-space.*

12. Russell, G. L. & Chernoff, E. J. (2013). The marginalization of Indigenous students within school mathematics and the math wars: seeking resolutions within ethical spaces [Special issue: Mathematics Education with/for Indigenous Peoples]. Mathematics Education Research Journal, 25(1), 109-127. doi: 10.1007/s13394-012-0064-1

Abstract.* In mathematics education, there are (at least) two seemingly disparate and unethical issues that have been allowed to continue unresolved for decades: the math wars (traditional versus reform teaching and learning of mathematics) and the marginalisation of Indigenous students within K-12 mathematics. Willie Ermine, an Indigenous scholar, has proposed the use of ethical spaces to explore and analyse occurrences of unethical situations arising between the “intersection of Indigenous law and Canadian Legal systems” (Ermine, Indigenous Law Journal 6(1):193–203, 2007). This paper brings Ermine’s notion of ethical spaces to the field of mathematics education research as the theoretical framework for analysing the aforementioned issues. The result of this analysis is a potential single theoretical resolution to both dilemmas that can also serve as a significant factor in the processes of decolonisation.*

11. Chernoff, E. J. (2012). Logically fallacious relative likelihood comparisons: the fallacy of composition [Special issue: National Year of Mathematics]. Experiments in Education, 40(4), 77-84.

Abstract. *The objective of this article is to contribute to research on prospective teachers’ probabilistic knowledge. To meet this objective, prospective mathematics teachers were presented with a novel task, which asked them to identify which result from five flips of a fair coin was least likely. However, unlike previous research, the participants were presented with events, that is, sets of outcomes, as opposed to sequences, which have dominated previous literature on relative likelihood comparisons. Recognizing that previous changes to the task have resulted in new areas of research, a new lens – the composition fallacy – was utilized when accounting for participants’ responses. Use of the new lens bolsters the contention that logical fallacies are a viable avenue for future investigations in comparisons of relative likelihood and research in probability.*

10. Chernoff, E. J. (2012). Recognizing revisitation of the representativeness heuristic: an analysis of answer key attributes [Themed issue: Probability in Reasoning About Data and Risk]. ZDM - The International Journal on Mathematics Education, 44(7), 941-952. doi: 10.1007/s11858-012-0435-9

Abstract.* The general objective of this article is to contribute to the limited research on teachers’ probabilistic knowledge. More specifically, this article aims to contribute to an established thread of research that investigates relative likelihood comparisons. To meet these objectives, prospective mathematics teachers were presented two different answer keys to a ten question multiple-choice quiz and were asked to determine and justify which of the two was least likely to occur. Unlike previous research, this article does not employ the representativeness heuristic, but, instead, utilizes the attribute substitution model—which stems from the genericism of the heuristics and biases program—to account for specific responses to relative likelihood comparisons. This new perspective demonstrates that certain individuals, when presented one question, answer a different question instead. Results demonstrate that participants substitute a variety of heuristic attributes instead of making the intended relative likelihood comparison of the answer keys presented.*

9. Chernoff, E. J., & Russell, G. L. (2012). The fallacy of composition: Prospective mathematics teachers’ use of logical fallacies. Canadian Journal of Science, Mathematics and Technology Education, 12(3), 259-271. doi: 10.1080/14926156.2012.704128

Abstract.* The purpose of this article is to address the lack of research on teachers’ knowledge of probability. As has been the case in prior research, we asked prospective mathematics teachers to determine which of the presented sequences of coin flips was least likely to occur. However, instead of using the traditional perspectives of heuristic and informal reasoning, we have utilized logical fallacies for our analysis of the results. From this new perspective, we determined that certain individuals’—those who provided normatively incorrect responses—utilized the fallacy of composition when making comparisons of relative likelihood. In addition, we discuss how our findings impact models established in the research literature (e.g., the representativeness heuristic) and, further, we suggest that logical fallacies should supplement heuristic and informal reasoning as potential perspectives for research investigating comparisons of relative likelihood.*

8. Chernoff, E. J. & Russell, G. L. (2011). The sample space: One of many ways to partition the set of all possible outcomes. The Australian Mathematics Teacher, 67(2), 24-29.

Abstract.* In this article, we discuss how acknowledging and embracing that the sample space is one of many ways to partition the set of all possible outcomes impacts the teaching and learning of sample space and proba- bility. After recounting an exchange surrounding two viable answers to a probability question, we detail how developments arising from mathematics education research investigating the partitioning of all possible outcomes can be integrated into the mathematics classroom. As a result, we present a unique perspective to normatively incorrect responses.*

7. Chernoff, E. J. & Zazkis, R. (2011). From personal to conventional probabilities: from sample set to sample space. Educational Studies in Mathematics, 77(1), 15-33. doi: 10.1007/s10649-010-9288-8

Abstract.* This article is a systematic reflection on a sequence of episodes related to teaching probability. Our central claim is that reducing problems to a consideration of the sample space, which consists of equiprobable outcomes, may not be in accord with learners’ initial ways of reasoning. We suggest a “desirable pedagogical approach” in which the solution builds on the set of outcomes as identified by learners and serves as a bridge towards mathematical convention. To explore prospective high school mathematics teachers’ ideas related to addressing a potential learner’s mistake and their reactions towards the suggested approach, we presented them with two tasks. In Task I, participants (n = 30) were asked to suggest a pedagogical remedy to a frequent mistake found in dealing with a standard probability problem, whereas in Task II, they were asked to solve a probabilistic problem, which they had not encountered previously. We discuss participants’ mathematical solutions to Task II in reference to their pedagogical approaches to Task I. The presented disparity serves in extending the convincing power of the suggested pedagogical approach.*

6. Russell, G., & Chernoff, E. J. (2011). Seeking more than nothing: Two elementary teachers’ conceptions of zero. The Montana Mathematics Enthusiast 8(1&2), 77-112.

Abstract.* Zero is a complex and important concept within mathematics, yet prior research has demonstrated that students, pre-service teachers, and teachers all have misconceptions about and/or lack of knowledge of zero. Using a hermeneutic approach based upon Gadamer’s philosophy, this study examined how two elementary mathematics teachers understand zero and how and when zero enters into their teaching of mathematics. The results of this study add new insights into the understandings of teachers and students related to zero and the origins, relationships between, and consequences of those understandings. Significant gaps and misconceptions within both teachers’ understandings of zero suggest the need for pre-service education programs to bring attention to the development of a more complete and meaningful understanding of zero.*

5. Chernoff, E. J. (2009). Sample space partitions: An investigative lens. Journal of Mathematical Behavior, 28(1), 19-29. doi: 10.1016/j.jmathb.2009.03.002

Abstract.* In this study subjects are presented with sequences of heads and tails, derived from flipping a fair coin, and asked to consider their chances of occurrence. In this new iteration of the comparative likelihood task, the ratio of heads to tails in all of the sequences is maintained. In order to help situate participants’ responses within conventional probability, this article employs unconventional set descriptions of the sample space organized according to: switches, longest run, and switches and longest run, which are all based upon subjects’ verbal descriptions of the sample space. Results show that normatively incorrect responses to the task are not devoid of correct probabilistic reasoning. The notion of alternative set descriptions is further developed, and the article contends that sample space partitions can act as an investigative lens for research on the comparative likelihood task, and probability education in general.*

4. Chernoff, E. J. (2008). The state of probability measurement in mathematics education: A first approximation. Philosophy of Mathematics Education Journal, 23, 1-23. Retrieved from http://people.exeter.ac.uk/PErnest/pome23/index.htm

Abstract.* In this article the three dominant philosophical interpretations of probability in mathematics education (classical, frequentist, and subjective) are critiqued. Probabilistic explorations of the debate over whether classical probability is belief-type or frequency-type probability will bring forth the notion that common ranges, rather than common points, of philosophical reference are inherent to probability measurement. In recognition of this point, refinement of subjective probability, into the dual classification of intrasubjective and intersubjective, and frequentist probability into the dual classification of artefactual and formal objective, attempts to address the nomenclatural issues inherent to subjective and frequentist probability being both general classifiers and particular theories. More specifically, adoption of artefactual and intersubjective probability will provide a more nuanced framework for the field to begin to heed the numerous calls put forth over the last twenty-five years for a unified approach to teaching and learning probability. Furthermore, the article proposes that “artefactual period” be adopted as a first approximation descriptor for the next phase of probability education.*

3. Zazkis, R., & Chernoff, E. (2008). What makes a counterexample exemplary? Educational Studies in Mathematics, 68(3), 195-208. doi: 10.1007/s10649-007-9110-4

Abstract.* In this paper we describe two episodes of instructional interaction, in which examples are used in order to help students face their misconceptions. We introduce the notions of pivotal example and bridging example and highlight their role in creating and resolving a cognitive conflict. We suggest that the convincing power of counterexamples depends on the extent to which they are in accord with individuals’ example spaces.*

2. Zazkis, R., Liljedahl, P. & Chernoff, E. (2008). The role of examples on forming and refuting generalizations [Themed issue: From Patterns to Generalization: Development of Algebraic Thinking]. ZDM - The International Journal on Mathematics Education, 40(1), 131-141. doi: 10.1007/s11858-007-0065-9

Abstract.* Acknowledging students’ difficulty in generalizing in general and expressing generality in particular, we assert that the choice of examples that learners are exposed to plays a crucial role in developing their ability to generalize. We share with the readers experiences in which examples supported generalization, and elucidate the strategies that worked for us in these circumstances, presuming that similar strategies could be helpful with other students in other settings. We further share several pitfalls and call for caution in avoiding them.*

1. Liljedahl, P., Chernoff. E., & Zazkis, R. (2007). Interweaving mathematics and pedagogy in task design: A tale of one task [Special issue: The Nature and Role of Tasks in Mathematics Teachers’ Education]. Journal of Mathematics Teacher Education, 10(4-6), 239-249. doi: 10.1007/s10857-007-9047-7

Abstract.* In this article we introduce a usage-goal framework within which task design can be guided and analyzed. We tell a tale of one task, the Pentomino Problem, and its evolution through predictive analysis, trial, reflective analysis, and adjustment. In describing several iterations of the task implementation, we focus on mathematical affordances embedded in the design and also briefly touch upon pedagogical affordances.*

**Articles in Non-Refereed Journals, Magazines and Periodicals (22)**

22. Chiasson, J. & Chernoff, E. J. (In Press). Finding Joy in the new math. The Variable: An SMTS Periodical, x(x), xx-xx.

21. Chernoff, E. J. (2019, Winter/Spring).The Canadian Math Wars: An Abridged History. The Variable, 4(1), 53-59.

20. Chernoff, E. J. (2018, November/December). Consumeracy: Consumer Numeracy. The Variable, 3(5), 46-50.

19. Chernoff, E. J. (2018, September/October). Little Signs of Innumeracy. The Variable, 3(4), 46-50.

18. Chernoff, E. J. (2018, May/June). Precisely Inaccurate: Putting the “Ass” in Assessment. The Variable, 3(3), 38-41.

17. Chernoff, E. J. (2018, March/April). Antiquated Arguments from Math Class: Calculating the Tip. The Variable, 3(2), 37-40.

16. Chernoff, E. J. (2018, January/February). A Spontaneous Celebration of Learning: Part I. [Math Ed Matters by MatthewMaddux]. The Variable: An SMTS Periodical, 3(1), 35-40.

15. Chernoff, E. J. (2017, November/December). The Inside Joke on Math Lessons. [Math Ed Matters by MatthewMaddux]. The Variable: An SMTS Periodical, 2(6), 38-42.

14. Chernoff, E. J. (2017, September/October). Abhorrent Mathematical Algorithms: Mathematical Abhorithms [Math Ed Matters by MatthewMaddux]. The Variable: An SMTS Periodical, 2(5), 44-50.

13. Vashchyshyn, I. & Chernoff, E. J. (2017, September). Sick of Viral Math. Math Horizons [Mathematical Association of America], 25(1), 33-34.

12. Chernoff, E. J. (2017, July/August). The Lottery is a Tax on... [Math Ed Matters by MatthewMaddux]. The Variable: An SMTS Periodical, 2(4), 39-48.

11. Chernoff, E. J. (2017, May/June). Subtraction: How the Hunted Became the Hunter [Math Ed Matters by MatthewMaddux]. The Variable: An SMTS Periodical, 2(3), 42-46.

10. Shaw, L. & Chernoff, E. J. (2017). A Gradeless Secondary Mathematics Classroom: Questions and Answers. Vector: Journal of the British Columbia Association of Mathematics Teachers, 58(1), 23-26.

9. Chernoff, E. J. (2015, October 26th). Lines You Can Count On [The Jersey Issue]. The Hockey News, 69(05), 11.

8. Prentice, C. & Chernoff, E. J. (2015). When two plus two equals blue: An overview of grapheme-color synesthesia research. Iowa Council of Teachers of Mathematics Journal, 41(Winter 2014-2015), 11-15.

7. Lehmkuhl, P. & Chernoff, E. J. (2014). The role of educational technology in relation to teacher and learner motivation in mathematics: a social perspective. Vector: Journal of the British Columbia Association of Mathematics Teachers, 55(3), 55-60.

6. Merkowsky, M. & Chernoff, E. J. (2014). The Absolutely True Confession of a Prospective Elementary School Math Teacher. Education Matters: The Journal of Teaching and Learning, 2(2), 41-52.

5. Chernoff, E. (2013). Two years and four issues later. vinculum: Journal of the Saskatchewan Mathematics Teachers' Society, 3(2.2000), 50-53. [Reprint of Chernoff, E. (2010). Editorial: Two years and four issues later. vinculum: Journal of the Saskatchewan Mathematics Teachers' Society, 2(2), 2-6.]

4. Chernoff, E. (2013). Change(s). vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.2000), 23. [Reprint of Chernoff, E. (2009). Editorial: Change(s). vinculum: Journal of the Saskatchewan Mathematics Teachers' Society, 1(1), 2.]

3. Chernoff, E. J. (2011). Where have all the submissions gone? Vector: Journal of the British Columbia Association of Mathematics Teachers, 52(3), 10-14.

2. Chernoff, E. J. (2010). Coming to terms with probability terminology. Vector: Journal of the British Columbia Association of Mathematics Teachers, 51(2), 13-16.

1. Chernoff, E. J. (2008). Now that’s what I call alternative base representation. Vector: Journal of the British Columbia Association of Mathematics Teachers, 49(2), 49-55.

**Encyclopedia Entries (3)**

3. Sriraman, B. & Chernoff, E. J. (in press). Probabilistic and Statistical Thinking. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer Reference. DOI: https://doi.org/10.1007/978-3-319-77487-9_100003-1

2. Chernoff, E. J. & Sriraman, B. (in press). Heuritics and Biases. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer Reference. DOI: https://doi.org/10.1007/978-3-319-77487-9_100010-1

1. Radakovich, N. & Chernoff, E. J. (in press). Risk Education. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer Reference. DOI: https://doi.org/10.1007/978-3-319-77487-9_100004-1

**Articles in Refereed Conference Proceedings (25)**

25. Chernoff, E. J., Vashchyshyn, I. & Neufeld, H. (2018). Comparing the relative probabilities of events. Proceedings of Topic Study Group 14: Teaching and learning of probability. 13th International Congress on Mathematics Education (ICME-13). Hamburg, Germany.[Available: https://iase-web.org/documents/papers/icme13/ICME13_I3_Chernoff.pdf]

24. Chernoff, E. J., & Russell, G. L. (2013). Comparing The Relative Likelihood Of Events: The Fallacy Of Composition. In Martinez, M. & Castro Superfine, A (Eds.), Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 333-340). Chicago, IL: University of Illinois at Chicago.

23. Russell, G. L., & Chernoff, E. J. (2013). Unifying challenges in the teaching and learning of mathematics: Two can become one. In Martinez, M. & Castro Superfine, A (Eds.), Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1018-1025). Chicago, IL: University of Illinois at Chicago.

22. Chernoff, E. J., & Russell, G. L. (2012). Why order does not matter: an appeal to ignorance. In Van Zoest, L. R., Lo, J.-J., & Kratky, J. L. (Eds.), Proceedings of the 34th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1045-1052). Kalamazoo, MI: Western Michigan University.

21. Russell, G. L., & Chernoff, E. J. (2012). Unifying challenges in the teaching and learning of mathematics: Two can become one. In Van Zoest, L. R., Lo, J.-J., & Kratky, J. L. (Eds.), Proceedings of the 34th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 367-370). Kalamazoo, MI: Western Michigan University.

20. Chernoff, E. J. (2012). Unintended relative likelihood comparisons. Proceedings of Topic Study Group 11: Teaching and learning of probability. 12th International Congress on Mathematics Education (ICME-12). Seoul, Korea.

19. Russell, G. L. & Chernoff, E. J. (2012). Unknown Occurrences of Polysemy in English Mathematics Classrooms. Proceedings of Topic Study Group 28: Language and communication in mathematics education. 12th International Congress on Mathematics Education (ICME-12). Seoul, Korea.

18. Chernoff, E. J. (2012). Providing answers to a question that was not asked. In S. Brown, S. Larsen, K. Marrongelle & M. Oehrtman (Eds.), Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education (pp. 32-38). Portland, Oregon.

17. Chernoff, E. J., & Russell, G. L. (2011). An informal fallacy in teachers’ reasoning about probability. 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. 241-249). Reno, NV: University of Nevada, Reno.

16. Russell, G. L., & Chernoff, E. J. (2011). Transforming mathematics education: applying new ideas or commodifying cultural knowledge. 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. 970-977). Reno, NV: University of Nevada, Reno.

15. Chernoff, E. J., & Russell, G. L. (2011). An investigation of relative likelihood comparisons: the composition fallacy. In B. Ubuz (Ed.), Proceedings of the Thirty fifth annual meeting of the International Group for the Psychology of Mathematics Education (Vol. II, pp. 225-232). Ankara, Turkey: Middle East Technical University.

14. Russell, G. L., & Chernoff, E. J. (2011). Logical fallacies in reasoning about a correct solution. In B. Ubuz (Ed.), Proceedings of the Thirty fifth annual meeting of the International Group for the Psychology of Mathematics Education (Vol. I, p. 379). Ankara, Turkey: Middle East Technical University.

13. Chernoff, E. J. (2011). Investigating relative likelihood comparisons of multinomial, contextual sequences. In M. Pytlak, T. Rowland, & E. Swoboda (Eds.), Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (pp. 755-765). University of Rzeszów, Poland.

12. Chernoff, E. J., & Zazkis, R. (2010). A problem with the problem of points. In P. Brosnan, D. Erchick, & L. Flevares (Eds.), Proceedings of the Thirty-Second Annual Meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (Vol. VI, pp. 969-977). Columbus, OH: Ohio State University.

11. Russell, G., & Chernoff, E. J. (2010). Beyond nothing: Teachers’ conceptions of zero. In P. Brosnan, D. Erchick, & L. Flevares (Eds.), Proceedings of the Thirty-Second Annual Meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (Vol. VI, pp. 1039-1046). Columbus, OH: Ohio State University.

10. Chernoff, E. J. (2009). The subjective-sample-space. In S. L. Swars, D. W. Stinson & S. Lemons-Smith (Eds.), Proceedings of the Thirty-First Annual Meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 628-635). Atlanta, GA: Georgia State University.

9. Chernoff, E. J. (2009). Explicating the multivalence of a probability task. In S. L. Swars, D. W. Stinson & S. Lemons-Smith (Eds.), Proceedings of the Thirty-First Annual Meeting of the North-American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 653-661). Atlanta, GA: Georgia State University.

8. Chernoff, E. J. (2008). Sample space: An investigative lens. In J. Cortina (Ed.), Proceedings of the Joint Meeting of the International Group and the North American Chapter for the Psychology of Mathematics Education (Vol. 2, pp. 313-320). Morelia, Michoacn, Mexico.

7. Chernoff, E. (2007). Sample space rearrangement (SSR): The example of switches and runs. In T. Lamberg & L. Wiest (Eds.), Proceedings of the Twenty Ninth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. (Vol. 1, pp. 433-436). Stateline (Lake Tahoe), NV: University of Nevada, Reno.

6. Chernoff, E. (2007). Probing Representativeness: Switches and runs, In J. Woo, H. Lew, and D. Seo (Eds.), Proceedings of the Thirty first annual meeting of the International Group for the Psychology of Mathematics Education. (Vol. 1, pp. 207). Seoul, South Korea: Seoul National University.

5. Chernoff, E. (2007). The Monistic Probabilistic Perspective. In J. Woo, H. Lew, and D. Seo (Eds.), Proceedings of the Thirty first annual meeting of the International Group for the Psychology of Mathematics Education.. (Vol. 1, pp. 308). Seoul, South Korea: Seoul National University.

4. Chernoff, E., & Zazkis, R. (2006). Intuitive probability in action: A case in elementary number theory. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the Twenty Eighth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. (Vol. 2, pp. 756-758). Mérida, Mexico: Universidad Pedagógica Nacional.

3. Zazkis, R., & Chernoff, E. (2006). Examples that change minds. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the Twenty Eighth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. (Vol. 2, pp. 756-758). Mérida, Mexico: Universidad Pedagógica Nacional.

2. Chernoff, E., & Zazkis, R. (2006). Decision making at uncertainty: Moving on a prime ladder. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the Thirtieth annual meeting of the International Group for the Psychology of Mathematics Education. (Vol. 1, pp. 234). Prague, Czech Republic: Charles University.

1. Zazkis, R., & Chernoff, E. (2006). Cognitive conflict and its resolution via pivotal/bridging example. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the Thirtieth annual meeting of the International Group for the Psychology of Mathematics Education. (Vol. 5, pp. 465-472). Prague, Czech Republic: Charles University.

**Articles in Non-Refereed Conference Proceedings (13)**

13. Russell, G. & Chernoff, E. J. (2018). The Problem with Deciding if Order Matters [Topic 6: Innovations in Teaching Probability]. Proceedings of the 10th International Conference of Teaching Statistics (ICOTS-10). Kyoto, Japan.

12. Larsen, J., Chernoff, E. J. & Freiman, V. (2018, May). Social Media for Mathematics Education. Working group C report for the Proceedings of the 41st annual meeting of the Canadian Mathematics Education Study Group/Groupe Canadien d'Étude en Didactique des Mathématiques. (pp. 77-90). Montreal, QC, Canada.

11. Batanero, C., Chernoff, E. J., Engel, J., S. Lee, H. & Sanchez, E. (2017). Topic Study Group 14: Description of Activities. In G. Kaiser (Ed.),Proceedings of the 13th International Congress on Mathematics Education (pp. 151-172). Hamburg, Germany: Springer.

10. Chernoff, E. J. (2017). The Canadian Journal of Science, Mathematics and Technology Education: Meet the editors. Ad-hoc presentation report for the proceedings of the 40th annual meeting of the Canadian Mathematics Education Study Group/Groupe Canadien d'Étude en Didactique des Mathématiques. (pp. 253-254).

9. Chernoff, E. (2014). Will the real Bayesian probability please stand up!? Proceedings of the 9th International Conference on Teaching Statistics (ICOTS9) [Session 6A: Bayesian inference (probability) goes to school: meanings, tasks and instructional challenges - Topic 6: Innovation and reform in teaching probability within statistics]. Flagstaff, Arizona, USA.

8. Chernoff, E. J. (2014). Social media and mathematics education: whenever the twain shall meet? In S. Oesterle & D. Allan (Eds.), Proceedings of the 2013 Annual Meeting of the Canadian Mathematics Education Study Group / Groupe Canadien d’Étude en Didactique des Mathématiques (pp. 143-147). St. Catharines, On: Brock University. CMESG/GCEDM.

7. Chernoff, E. J. (2011). Mathematics education networking experiences: The necessary, the unnecessary, and the digital. Proceedings of the Third Annual Mathematics Education Graduate Students’ Association (MEGA) Conference and Meeting. Vancouver, Canada: University of British Columbia. [Online: http://m1.cust.educ.ubc.ca/mega2011/proceedings.html]

6. Chernoff, E. J. (2011). Subjective probabilities derived from the perceived randomness of sequences of outcomes. New PhD report for the proceedings the 34th annual meeting of the Canadian Mathematics Education Study Group/Groupe Canadien d'Étude en Didactique des Mathématiques. (pp. 165-170). Vancouver, Canada: Simon Fraser University.

5. Chernoff, E. J., Knoll, E., & Mamolo, A. (2011). Noticing and engaging the mathematicians in our classrooms. Working group F report for the Proceedings of the 34th annual meeting of the Canadian Mathematics Education Study Group/Groupe Canadien d'Étude en Didactique des Mathématiques. (pp. 107-120). Vancouver, Canada: Simon Fraser University.

4. Chernoff, E. J., Chorney, S., & Liljedahl, P. (2011). Editing mathematics teachers’ journals in Canada: Bridging the gap between researchers and teachers. Ad-hoc presentation report for the proceedings of the 34th annual meeting of the Canadian Mathematics Education Study Group/Groupe Canadien d'Étude en Didactique des Mathématiques. (pp. 217-218). Vancouver, Canada: Simon Fraser University.

3. Chernoff, E. J. (2009). Panel I Report: What did I need then? What do I need now? Proceedings of the 2009 Canadian Mathematics Education Forum. Vancouver, Canada.

2. Chernoff, E. J. (2009). The Kamloops Golf and University Country Club. In R. C. Brewster, & J. G. McLoughlin (Eds.), Proceedings of the first annual Sharing Mathematics: A Tribute to Jim Totten conference. (pp. 86-87). Kamloops, British Columbia, Canada.

1. Chernoff, E., & Savard, A. (2008). Probability. Proceedings of the 2007 Annual Meeting of the Canadian Mathematics Education Study Group/Groupe Canadien d'Étude en Didactique des Mathématiques.

**Books and Monographs (7)**

7. Chernoff, E. J., Russell, G. L., & Sriraman, B. (Eds.) (in press). *Selected writings from the Journal of the Saskatchewan Mathematics Teachers’ Society: Celebrating 50 years (1961-2011) of vinculum*. Charlotte, NC: Information Age Publishing.

6. Kajander, A., Holm, J. & Chernoff, E. J. (Eds.) (2018). Teaching and Learning Secondary School Mathematics: Canadian Perspectives in an International Context [Advances in Mathematics Education Series]. Berlin Heidelberg: Springer Nature. (695 pages.)

5. Batanero, C. & Chernoff, E. J. (Eds.) (2018). Teaching and Learning Stochastics: Advances in Probability Education Research [ICME-13 Monograph Series]. Berlin/Heidelberg: Springer Nature.

4. Batanero, C., Chernoff, E. J., Engel, J., S. Lee, H. & Sanchez, E. (2016). Essential Research on Teaching and Learning Probability [ICME-13 Topical Surveys Series]. Springer. [40 pages.]

3. Chernoff, E. J., Liljedahl, P., & Chorney, S. (Eds.) (2016). *Selected writings from the Journal of the British Columbia Association of Mathematics Teachers: Celebrating 50 (1962-2012) years of Vector*. Charlotte, NC: Information Age Publishing. (443 pages.)

2. Chernoff, E. J. & Sterenberg, G. (Eds.) (2014). *Selected writings from the Journal of the Mathematics Council of the Alberta Teachers’ Association: Celebrating 50 years (1962-2012) of delta-K.* Charlotte, NC: Information Age Publishing. (500 pages.)

1. Chernoff, E. J., & Sriraman, B. (Eds.) (2014). Probabilistic Thinking: Presenting Plural Perspectives (Volume 7 of Advances in Mathematics Education Series). Berlin/Heidelberg: Springer Science. (748 pages.)

0. Chernoff, E. J. (2009). The subjective-sample-space: Subjective probabilities derived from the perceived randomness of sequences of outcomes*. Saarbrücken, Germany: VDM Verlag.

*note: this monograph is a verbatim reprint of my doctoral dissertation

**Book and Monograph Matter (Forewords, Prefaces, Introductions, Commentaries) (22)**

**Forewords and Prefaces (6)**

7. Kajander, A., Chernoff, E. J. & Holm, J. (2018). Preface. In A. Kajander, J. Holm & E. J. Chernoff (Eds.), Teaching and Learning Secondary School Mathematics: Canadian Perspectives in an International Context (pp. xii-xiv). Berlin/Heidelberg: Springer Science.

6. Batanero, C. & Chernoff, E. J. (2018). Preface. In C. Batanero & E. J. Chernoff (Eds.), Teaching and Learning Stochastics: Advances in Probability Education Research (pp. v-viii). Berlin/Heidelberg: Springer Science.

5. Chernoff, E. J., Chorney, S., & Liljedahl, P. (2016). Preface. In E. J. Chernoff, P. Liljedahl, & S. Chorney (Eds.), Selected writings from the Journal of the British Columbia Association of Mathematics Teachers: Celebrating 50 years (1962-2012) of Vector (pp. xvii-xxiv). Charlotte, NC: Information Age Publishing.

4. Chernoff, E. J. & Sterenberg, G. (2014). Preface. In E. J. Chernoff & G. Sterenberg (Eds.), Selected writings from the Journal of the Mathematics Council of the Alberta Teachers’ Association: Celebrating 50 years (1962-2012) of delta-K (pp. xix-xxiii). Charlotte, NC: Information Age Publishing.

3. Chernoff, E. J., & Russell, G. L. (2014). Preface to Perspective I: Mathematics and Philosophy. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic Thinking: Presenting Plural Perspectives (pp. 3-6). Berlin/Heidelberg: Springer Science.

2. Chernoff, E. J., & Russell, G. L. (2014). Preface to Perspective III: Stochastics. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic Thinking: Presenting Plural Perspectives (pp. 343-344). Berlin/Heidelberg: Springer Science.

1. Chernoff, E. J., & Russell, G. L. (2014). Preface to Perspective IV: Mathematics Education. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic Thinking: Presenting Plural Perspectives (pp. 493-494). Berlin/Heidelberg: Springer Science.

**Introductions (8)**

8. Chernoff, E. J. (in press). Introduction: The Eighties. In E. J. Chernoff, G. L. Russell, & B. Sriraman (Eds.), Selected writings from the Journal of the Saskatchewan Mathematics Teachers’ Society: Celebrating 50 years (1961-2011) of vinculum. Charlotte, NC: Information Age Publishing.

7. Chernoff, E. J. (in press). Introduction: The Nineties. In E. J. Chernoff, G. L. Russell, & B. Sriraman (Eds.), Selected writings from the Journal of the Saskatchewan Mathematics Teachers’ Society: Celebrating 50 years (1961-2011) of vinculum. Charlotte, NC: Information Age Publishing.

6. Chernoff, E. J. (in press). Introduction: The Aughts. In E. J. Chernoff, G. L. Russell, & B. Sriraman (Eds.), Selected writings from the Journal of the Saskatchewan Mathematics Teachers’ Society: Celebrating 50 years (1961-2011) of vinculum. Charlotte, NC: Information Age Publishing.

5. Russell, G. L. & Chernoff, E. J. (in press). Introduction: The Sixties. In E. J. Chernoff, G. L. Russell, & B. Sriraman (Eds.), Selected writings from the Journal of the Saskatchewan Mathematics Teachers’ Society: Celebrating 50 years (1961-2011) of vinculum. Charlotte, NC: Information Age Publishing.

4. Chernoff, E. J. (in press). Introduction: The Seventies. In E. J. Chernoff, G. L. Russell, & B. Sriraman (Eds.), Selected writings from the Journal of the Saskatchewan Mathematics Teachers’ Society: Celebrating 50 years (1961-2011) of vinculum. Charlotte, NC: Information Age Publishing.

3. Chernoff, E. J., Russell, G. L., & Sriraman, B. (in press). Introduction. In E. J. Chernoff, G. L. Russell, & B. Sriraman (Eds.), Selected writings from the Journal of the Saskatchewan Mathematics Teachers’ Society: Celebrating 50 years (1961-2011) of vinculum. Charlotte, NC: Information Age Publishing.

2. Sterenberg, G., & Chernoff, E. J. (2014). Introduction. In E. J. Chernoff & G. Sterenberg (Eds.), Selected writings from the Journal of the Mathematics Council of the Alberta Teachers’ Association: Celebrating 50 years (1962-2012) of delta-K (pp. xxv-xxxiii). Charlotte, NC: Information Age Publishing.

1. Chernoff, E. J., & Sriraman, B. (2014). Introduction to Probabilistic Thinking: Presenting Plural Perspectives. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic Thinking: Presenting Plural Perspectives (pp. xv-xviii). Berlin/Heidelberg: Springer Science.

**Commentaries (7)**

7. Russell, G. L. & Chernoff, E. J. (in press). Commentary: The Seventies. In E. J. Chernoff, G. L. Russell, & B. Sriraman (Eds.), Selected writings from the Journal of the Saskatchewan Mathematics Teachers’ Society: Celebrating 50 years (1961-2011) of vinculum. Charlotte, NC: Information Age Publishing.

6. Chernoff, E. J., Russell, G. L., & Sriraman, B. (in press). A Final Commentary. In E. J. Chernoff, G. L. Russell, & B. Sriraman (Eds.), Selected writings from the Journal of the Saskatchewan Mathematics Teachers’ Society: Celebrating 50 years (1961-2011) of vinculum. Charlotte, NC: Information Age Publishing.

5. Kajander, A., Holm, J. & Chernoff, E. J. (2018). Final Commentary. In A. Kajander, J. Holm & E. J. Chernoff (Eds.), Teaching and Learning Secondary School Mathematics: Canadian Perspectives in an International Context (pp. 641-647). Berlin/Heidelberg: Springer Nature.

4. Chernoff, E. J. & Batanero, C. (2018). A Commentary on Teaching and Learning Stochastics: Advances in Probability Education Research. In C. Batanero & E. J. Chernoff (Eds.), Teaching and Learning Stochastics: Advances in Probability Education Research (pp. 373-376). Berlin/Heidelberg: Springer Nature.

3. Chorney, S., Liljedahl, P. & Chernoff, E. J. (2016). Final Commentary: Looking back on our selected writings from fifty years of Vector. In E. J. Chernoff, P. Liljedahl, & S. Chorney (Eds.), Selected writings from the Journal of the British Columbia Association of Mathematics Teachers: Celebrating 50 years (1962-2012) of Vector (pp. 438-439). Charlotte, NC: Information Age Publishing.

2. Chernoff, E. J., & Sterenberg, G. (2014). Final commentary: Looking back on our selected writings from fifty years of delta-K. In E. J. Chernoff & G. Sterenberg (Eds.), Selected writings from the Journal of the Mathematics Council of the Alberta Teachers’ Association: Celebrating 50 years (1962-2012) of delta-K (pp. 459-466). Charlotte, NC: Information Age Publishing.

1. Chernoff, E. J., & Sriraman, B. (2014). Commentary on Probabilistic Thinking: Presenting Plural Perspectives. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic Thinking: Presenting Plural Perspectives (pp. 721-728). Berlin/Heidelberg: Springer Science.

**Chapters in Edited Books (5)**

5. Chernoff, E. J. (In Press). Sample Sapce Subjectivism. In V. Martin, M. Thibault & L. Theis (Eds.), Enseigner les premiers concepts de probabilités : un monde de possibilités! Presses de l’Université du Québec.

4. Banting, N. Vashchyshyn, I. & Chernoff, E. J. (2018). In No Uncertain Terms: Encouraging a Critical Stance Toward Probability in School. In A. Kajander, J. Holm & E. J. Chernoff (Eds.), Teaching and Learning Secondary School Mathematics: Canadian Perspectives in an International Context (pp. 571-588). Berlin/Heidelberg: Springer Nature.

3. Chernoff, E. J., Vashchyshyn, I. & Neufeld, H. (2018). Comparing the relative probabilities of events. In C. Batanero & E. J. Chernoff (Eds.), Teaching and Learning Stochastics: Advances in Probability Education Research (pp. 277-292). Berlin/Heidelberg: Springer Science.

2. Vashchyshyn, I. & Chernoff, E. J. (2018). Obstacles to a Transdisciplinary Resolution of the Math Wars. In L. Jao & N. Radakovic (Eds.), Transdisciplinarity in Mathematics Education: Blurring Disciplinary Boundaries (151-172). Springer.

Abstract.

*Faced with the complex issues of modern society, a growing number of individuals and organisations have embraced a transdisciplinary approach in the attempt to resolve such issues in an ethical, socially responsible way. Such an approach may even prove to be effective in mediating (if not resolving) the math wars, a longstanding, value-laden debate about what (mathematics) children should learn in the twenty-first century and how they should learn it. However, although the math wars have evolved into a conflict involving a wide variety of individuals and groups representing various interests and disciplines, we argue that for this issue, transdisciplinarity is still out of reach. In particular, in re-viewing the evolution of the math wars in the United States and in Canada through a transdisciplinary lens, we find that one major obstacle is the reluctance, and sometimes outright refusal, to step outside disciplinary constraints to engage in dialogue and collaboration with diverse stakeholders. We contend that if the attitude of opposition is maintained, we should expect a long and bitter war indeed.*

1. Chernoff, E. J. & Sriraman, B. (2015). The teaching and learning of probabilistic thinking: heuristic, informal and fallacious reasoning. In R. Wegerif, L. Li & J. Kaufman (Eds.), The Routledge International Handbook of Research on Teaching Thinking (pp. 369-377). New York: Routledge, Taylor & Francis.

**Edited Volumes (13)**

13. Chernoff, E. J., Paparistodemou, E., Bakogianni, D., & Petocz, P. (Guest Editors) (2016). Special Issue: Research on learning and teaching probability within statistics. Statistics Education Research Journal, 15(2). 265 pages.

12. Chernoff, E. J. (Guest Editor) (2015). Special Issue: Risk-Mathematical or Otherwise. The Mathematics Enthusiast, 12(1,2&3). 479 pages.

11. Chernoff, E. (Ed.) (2014). Special Issue: The Collected Works of Rick Seaman. vinculum: Journal of the Saskatchewan Mathematics Teachers' Society, 4(1&2). 96 pages.

10. Chernoff, E. (Ed.) (2013). Celebrating 50 years (1961-2011) of the SMTS: The Aughts. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.2000). 53 pages.

9. Chernoff, E. (Ed.) (2013). Celebrating 50 years (1961-2011) of the SMTS: The Nineties. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.1990). 63 pages.

8. Chernoff, E. (Ed.) (2012). Celebrating 50 years (1961-2011) of the SMTS: The Eighties. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.1980). 56 pages.

7. Chernoff, E. (Ed.) (2012). Celebrating 50 years (1961-2011) of the SMTS: The Seventies. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.1970). 64 pages.

6. Chernoff, E. (Ed.) (2012). Celebrating 50 years (1961-2011) of the SMTS: The Sixties. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.1960). 49 pages.

5. Chernoff, E. (Ed.) (2011). Theme: Problems and reflections. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(1). 56 pages.

4. Chernoff, E. (Ed.) (2010). Theme: First Nations and Métis content, perspectives, and ways of knowing. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 2(2). 68 pages.

3. Chernoff, E. (Ed.) (2010). Theme: Curricular edition. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 2(1). 60 pages.

2. Chernoff, E. (Ed.) (2009). Theme: Student-centered edition. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 1(2). 52 pages.

1. Chernoff, E. (Ed.) (2009). vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 1(1). 44 pages.

**Editorials (14)**

14. Chernoff, E. J., Paparistodemou, E., Bakogianni, D., & Petocz, P. (2016) Guest Editorial: Working Title. Special Issue: Research on learning and teaching probability within statistics. Statistics Education Research Journal, 15(2), 6-10.

13. Chernoff, E. J. (2015) Guest Editorial: Risk-Mathematical or Otherwise. The Mathematics Enthusiast, 12(1,2&3), 3.

12. Chernoff, E. (2014). Editorial: Change(s), five years later. vinculum: Journal of the Saskatchewan Mathematics Teachers' Society, 4(1&2), 1.

11. Chernoff, E. (2013). Editorial: the aughts. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.2000), 1.

10. Chernoff, E. (2013). Editorial: the nineties. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.1990), 1.

9. Chernoff, E. (2012). Editorial: the eighties. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.1980), 1.

8. Chernoff, E. (2012). Editorial: the seventies. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.1970), 1.

7. Russell, G. & Chernoff, E. (2012). Editorial: The sixties: "The times they are a-changin". vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.1960), 4.

6. Chernoff, E. (2012). Preface: celebrating 50 years (1961-2011) of the SMTS. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(2.1960), 2-3.

5. Chernoff, E. (2011). Editorial: No, not that kind of problem. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 3(1), 3-4.

4. Chernoff, E. (2010). Editorial: Two years and four issues later. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 2(2), 2-6.

3. Chernoff, E. (2010). Editorial: Curricular edition. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 2(1), 2-3.

2. Chernoff, E. (2009). Editorial: Student-centered edition. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 1(2), 3-4.

1. Chernoff, E. (2009). Editorial: Change(s). vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 1(1), 2.

**Dissertation (1)**

Chernoff, E. J. (2009). Subjective probabilities derived from the perceived randomness of sequences of outcomes. Unpublished doctoral dissertation. Simon Fraser University, Vancouver, British Columbia, Canada.

**eBooks (3)**

3. Chernoff, E. J. (2012). *Mathematics Education Images by MatthewMaddux.* Available on iTunes (36 pages).

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2. Chernoff, E. J. (2012). *@MatthewMaddux 2011: Chronicled Tweet by Tweet*. Available on iTunes (182 pages).

1. Chernoff, E. J. (2012). *MatthewMaddux Education 2011.* Retrieved from http://www.eganchernoff.com/downloads/open-access-publications/ (220 pages).

**Book Reviews (1)**

1. Chernoff, E. (2009). Innumeracy: Mathematical illiteracy and its consequences: A review. vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society, 1(1), 36-41.

**Other Publications (5)**

5. Chernoff, E. J. (2018, September 25). Leaving Rutherford for Anotherford: While Rutherford Rink had its challenges, for the Faculty/Staff Hockey League, Rutherford's problems were sometimes a blessing in disguise. Alumni and Friends, University of Saskatchewan, Saskatoon, SK, Canada. Available here: https://alumni.usask.ca/news/2018/remembering-rutherford-rink-an-fshlers-perspective.php

4. Chernoff, E.J. (2016). Yukongradulations! In, 50 Stories | 1965-2015. Simon Fraser University.

3. Chernoff, E. J. (2010). In memoriam: Craig Newell.

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