Abstracts of Selected Publications
Lesser, L. (2014; in press). Mathematical lyrics: Noteworthy endeavors in education. Journal of Mathematics and the Arts
Abstract: Mathematical lyrics are song lyrics connected to, or inspired by, mathematics or statistics. This paper explores various types of mathematical lyrics and their roles in mathematics education. In particular, the paper contains many examples of my own lyrics as well as an extensive bibliography of lyrics composed by others. It also provides resources and strategies for creating such lyrics and for using them in an educational setting.
* Lesser, L., Wagler, A. & Abormegah, P. (2014). Finding a happy median: Another balance representation for measures of center, Journal of Statistics Education, 22(3), 1-27. http://www.amstat.org/publications/jse/v22n3/lesser.pdf
Abstract: This paper explores the use of a lesser-known dynamic model for the median, a foundational
topic that starts in the middle school curriculum and is associated with student misconceptions
and knowledge gaps. This model appears to offer a rich vehicle to explore the median
interactively in greater conceptual depth that includes some of its more subtle associated ideas.
An exploratory study to assess performance of this model in a class for pre-service middle school
teachers yielded evidence that students who completed the dataset sequence associated with the
model gained further insight about the median, especially concerning how the mean and median
are affected differently by outliers. Analyses of open ended questions as well as empirical results
of multiple-choice questions are used to assess the overall learning outcomes gained by students.
* Harrell-Williams, L.; Sorto, M. A.; Pierce, R. L.; Lesser, L. M.; Murphy, T. J. (2014). Validation of scores from a new measure of pre-service teachers’ self-efficacy to teach statistics in the middle grades. Journal of Psychoeducational Assessment, 32(1), 40-50.
Abstract: The influential Common Core State Standards for Mathematics (CCSSM) expect students to start statistics learning during middle grades. Thus, teacher education and professional development programs are advised to help pre-service and inservice teachers increase their knowledge and confidence to teach statistics. Although existing self-efficacy instruments used in statistics education focus on students, the Self-Efficacy to Teach Statistics (SETS) instrument measures a teacher’s efficacy to teach key CCSSM statistical topics. Using the results from a sample of n = 309 participants enrolled in a mathematics education or introductory statistics course, SETS scores were validated for use with middle grades pre-service teachers to differentiate levels of self-efficacy to teach statistics. Confirmatory factor analysis using the Multidimensional Random Coefficient Multinomial Logit Model supports the use of two dimensions, which exhibit adequate reliabilities and correspond to the first two levels of The Guidelines for Assessment and Instruction in Statistics Education adopted by the American Statistical Association. Item and rating scale analyses indicate that the items and the six-category scale perform as intended. These indicators suggest that the SETS instrument may be appropriate for measuring pre-service teacher levels of self-efficacy to teach statistics.
* Lesser, L., Wagler, A., Esquinca, A., & Valenzuela, M. G. (2013). Survey of native English speakers and Spanish-speaking English language learners in tertiary introductory statistics. Statistics Education Research Journal, 12(2), 6-31. Abstract: The framework of linguistic register and case study research on Spanish-speaking English language learners (ELLs) learning statistics informed the construction of a quantitative instrument, the Communication, Language And Statistics Survey (CLASS). CLASS aims to assess whether ELLs and non-ELLs approach the learning of statistics differently with respect to the distinctive linguistic features of the field of statistics and with respect to language resources they bring to the class. The CLASS was administered to all (n = 137) students in an introductory statistics literacy course at a university with a majority Mexican-American student body. Findings suggest ELLs often have distinctive patterns in how they experience aspects of statistics instruction (e.g., wait time) as well as movement between mathematics/statistics and everyday registers.
* Lesser, L. (2013). Types of average and sampling: ‘Household words’ to dwell on. Statistics Education Web: Online Journal of K-12 Statistics Lesson Plans (pp. 1-12). [Bronze medal prize-winner in contest sponsored by ASA] http://www.amstat.org/education/stew/pdfs/TypesofAverageandSamplingHouseholdWordstoDwellOn.pdf
Abstract: This lesson is designed to give students more insight into both mathematical and real-world assumptions that can be involved even in what appears to be a simple task: to find the “average household size” for students at their school. The context of this lesson has real-world relevance: it is not only part of the Census at School questionnaire, but is also part of what the Census Bureau regularly estimates. The lesson can efficiently set the tone for better habits of mind and questioning and more precise usage of vocabulary and more explicit awareness of assumptions and possible biases in a broad range of future student work in statistics.
* Lesser, L.M.; Wall, A.; Carver, R.; Pearl, D.K.; Martin, N.; Kuiper, S.; Posner, M. A.; Erickson, P.; Liao, S.-M.; Albert, J.; Weber, J.J. (2013). Using fun in the statistics classroom: An exploratory study of college instructors’ hesitations and motivations. Journal of Statistics Education, 21(1), 1-33. http://www.amstat.org/publications/jse/v21n1/lesser.pdf
Abstract: This study examines statistics instructors’ use of fun as well as their motivations, hesitations, and awareness of resources. In 2011, a survey was administered to attendees at a national statistics education conference, and follow-up qualitative interviews were conducted with 16 of those (N = 249) surveyed to provide further context for interpreting the quantitative results. Motivations were similar for men and women, but female instructors admitted more hesitations in many areas. While many instructors are using or are open to using fun in the statistics classroom, the findings suggest that not having available resources at hand and not being aware of resources such as the CAUSEweb collection are major hesitations. Methods of alleviating hesitations are discussed.
*Lesser, L. (2013). Using context to classify variables. Teaching Statistics, 35(1), 29-31. http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9639.2012.00527.x/pdf
Asking clarifying questions and utilizing context can help students with several aspects of introductory statistics, such as appropriately classifying variables.
*Lesser, L. & Guthrie, J. (2012). Uncommon Connections with Common Numerators. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 22(8), 634-642. Abstract: Undergraduate students who are pre-service teachers need to make connections between the college mathematics they are learning and the precollege mathematics they will be teaching. Spanning a broad range of undergraduate curricula, this article describes useful lesser-known connections, explorations and original new proofs involving fractions. In particular, we use standard tools of number theory to investigate when an integer results from bc/(b+c), an expression that arises from the not necessarily uncommon situation of adding fractions with a common numerator.
* Rubel, L., Driskill, M., and Lesser, L. (October 2012). Decennial Redistricting: Rich Mathematics in Context. Mathematics Teacher, 106(3), 206-211. http://www.nctm.org/publications/article.aspx?id=34455
Abstract: Decennial redistricting is a real-world context rich in mathematics accessible to high school students. This article provides teachers an overview about redistricting with resources and examples of its diverse mathematical connections.
* Lesser, L. (2012). High-Speed Hypotheses! Teaching Statistics, 34(1), 10-12.
Abstract: This article presents engaging interactive hypothesis tests which can be conducted with students very efficiently.
*Lesser, L. (2011). Low-Tech, Low-Cost, High-Gain, Real-Time Assessment? It’s all in the cards, easy as ABCD! Texas Mathematics Teacher, 58(2), 18-22. http://www.math.utep.edu/Faculty/lesser/LesserABCDcardsTMTpaper.pdf or at http://tctmonline.org/TMT_archive.html. Abstract: An accessible overview of method, rationale, and references for using classroom voting cards in the college statistics classroom, including concrete examples and data collected to assess the innovation.
*Lesser, L. M., & Kephart, K. (Nov. 2011). Setting the tone: A discursive case study of problem-based inquiry learning to start a graduate statistics course for in-service teachers. J. of Statistics Education, 19(3), 1-29. http://www.amstat.org/publications/jse/v19n3/lesser.pdf [This paper won the Scholarship of Teaching and Learning Award at the 2012 International Sun Conference on Teaching and Learning]
Abstract: The first day of a course has great potential to set the tone for the entire course, planting the seeds for habits of mind and questioning and setting in motion expectations for classroom discourse. Rather than let the first meeting contain little besides going over the syllabus, the instructor (Lesser) decided to use two sustained open-ended scenarios to put in place from the start the problem-based inquiry learning approach he wanted to use throughout most of the course. After reviewing the literatures involved, this paper shares a description of the lesson‘s design and instructional cycle and a discourse analysis of that lesson‘s implementation. Strategies identified by the case study analysis include varying participation structures, well-crafted problems, and the instructor‘s role as facilitator and co-learner.
*Wagler, A. E., and Lesser, L. M. (2011). Teaching statistics to culturally and linguistically diverse students. Proceedings of the 2011 Joint Statistical Meetings, Section on Statistical Education (pp. 821-830). http://www.statlit.org/pdf/2011Wagler-Lesser-JSM.pdf or https://www.amstat.org/membersonly/proceedings/2011/papers/300678_65313.pdf Abstract: Resources, strategies, and lessons learned are shared based on the authors’ research and
experience in teaching statistics (especially introductory statistics) to diverse student populations in varied settings.
*Lesser, L. (2011). On the use of mnemonics for teaching statistics. Model Assisted Statistics and Applications, 6(2), 151-160.
*Lesser, L. M. (2011) Making statistics memorable: New mnemonics and motivations. Proceedings of the 2011 Joint Statistical Meetings, Section on Statistical Education (pp. 1118-1124). https://www.amstat.org/membersonly/proceedings/2011/papers/300805_65461.pdf or http://www.statlit.org/pdf/2011Lesser-JSM.pdf
Abstract: This paper is a comprehensive attempt to compile and classify mnemonics (memory aids) that can be used in statistics education.
*Sorto, M.A., White, A., and Lesser, L. (2011) Understanding Student Attempts to Find a Line of Fit. Teaching Statistics, 33(2), 49-52.
Abstract: The least squares method of fitting a line is not one that naturally occurs to students. We present three tasks to understand student views on how lines may be fit.
*Lesser, L. (2011). Supporting Learners of Varying Levels of English Proficiency. Statistics Teacher Network, 77, 2-5.
Abstract: Ten concrete research-informed recommendations are given for those who teach statistics to K-12 students with varying levels of English proficiency.
* Lesser, L. (2011). Simple Datasets for Distinct Basic Summary Statistics. Teaching Statistics, 33(1), 9-11.
Abstract: Datasets with simple numbers that lead to distinct basic summary statistics and preserve certain types of generality are provided, accompanied by criteria and rationale. Connections to the education literature are made and suggestions are made for classroom use.
*Lesser, L. (2010). The Necessity of Equity in Teaching Statistics. Philosophy of Mathematics Education Journal, no. 25, http://people.exeter.ac.uk/PErnest/pome25/index.html, 8 pp.
Abstract: Students have prior conceptions about fairness, and these must be acknowledged by teachers of statistics because these prior conceptions may interact or interfere with learning or embracing statistics concepts such as expected value, random sampling, random assignment, bias, and discrimination. To the extent equity includes not just fairness, but also matters of culture, it may also be worth considering how culture can interfere with the interpretation of a common metaphor for hypothesis testing.
* Lesser, L. (2010). Equity and the increasingly diverse tertiary student population: challenges and opportunities in statistics education. Proceedings of the 8th International Conference on Teaching Statistics (6 pp.).
Abstract: Professional organizations and educators are increasing their focus on equity issues for the diverse (sub)populations they serve, such as second language learners. This focus is especially important in statistics because of the importance of context in both diversity and statistics, and because of the ways in which diversity issues interact with the actual practice and teaching of statistics. Also, there are many ways in which diversity and equity can be natural vehicles for motivating or exploring particular statistical concepts and content. Recently emerging evidence suggests that students in diverse classrooms can be motivated by such examples and there have been a variety of courses that have recently been developed to do this with varying emphases and degrees of integration. The totality of efforts in this area suggests that the challenges of teaching diverse populations ultimately yield opportunities, resources, and vehicles for student learning.
* Lesser, L. (2010). An Ethnomathematics Spin on Statistics Class. Notices of the North American Study Group in Ethnomathematics (NASGEm News), 3(2), 5-6. http://nasgem.rpi.edu/files/2055/
Abstract: Ethnomathematics can also be applied to probability and statistics, and one concrete example that was explored by the author with a predominantly Mexican-American college student population is the Mexican game of Toma Todo. Connections are made to ethnomathematics/research literature, pedagogical issues, and related multicultural games.
* Sorto, M.
Alejandra & Lesser,
* Lesser, L. (Dec. 2009/Jan. 2010). Sizing Up Class Size: A Deeper Classroom Investigation of Central Tendency. Mathematics Teacher, 103(5), 376-380. Abstract: A common real-world question about “average class size” yields a surprisingly rich exploration of conceptual and procedural knowledge about measures of location. Pedagogical connections are made to the role of simple numbers (Lesser and Melgoza 2007), the role of assumptions, algebra-based deductive reasoning, real-world context, and the inspection paradox. [note: The exploration was classroom tested as a survey of (N= 50) pre-service elementary and middle school teachers in two sections of a required introductory statistics course at a mid-sized doctoral research intensive university in the Southwestern United States. The pre-service teachers uniformly focused on the simplest interpretation of mean and were generally surprised to see how many other interpretations were possible.]
* Lesser, L. M. & Glickman, M. E. (Dec. 2009). Using Magic in the Teaching of Probability and Statistics. Model Assisted Statistics and Applications, 4(4), 265-274. Abstract: This paper explores the role magic tricks can play in the teaching of probability and statistics, especially for lectures in college courses. Demonstrations are described that illustrate a variety of probabilistic and statistical topics, including basic probability and combinatorics, probability and sampling distributions, hypothesis testing, and advanced topics such as Markov chains and Bayes' Theorem. In addition to magic tricks providing visual demonstrations to supplement traditional blackboard-based lectures and the opportunity to engage students in class-participatory activities, possible benefits include a focus on conceptual understanding, development of critical thinking, and an opportunity to reflect upon the role of assumptions and estimates of probabilities.
* Lesser, L. & Winsor, M. (Nov. 2009), English Language Learners in Introductory Statistics: Lessons Learned from an Exploratory Case Study of Two Pre-Service Teachers. Statistics Education Research Journal, 8(2), 5-32. Abstract: Despite the rapidly growing population of English language learners in US colleges and schools, very little research has focused on understanding the challenges of English language learners specifically in statistics education. At a university near the United States-México border, the authors conducted an exploratory qualitative case study of issues of language in learning statistics for pre-service teachers whose first (and stronger) language is Spanish. The two strongest findings that emerged from cross-case analysis of the interviews were the importance of the role of context (the setting in which information is communicated) and the confusion between registers (subsets of language). This paper overviews and synthesizes relevant literature and offers resources and recommendations for teaching and future research.
* Matthew Winsor & Larry Lesser (November 2009) Hot Wheels: A Vehicle for Proportional Reasoning. Mathematics Teaching in the Middle School, 15(4), 239-243. Abstract: This article describes an activity that uses Hot Wheels cars to examine concepts of scale and proportionality. Student thinking and extensions to other real world applications are also addressed.
Lesser, L. & Groth, R. (2009). Technological Pedagogical Content Knowledge in Statistics. In Joanne Foster (Ed.), CD-ROM Proceedings of the Twentieth Annual International Conference on Technology in Collegiate Mathematics, pp. 148-152. Boston: Pearson. ISBN 0-321-64488-3 [Reprint available in the Electronic Proceedings of the 20th Annual ICTCM: http://archives.math.utk.edu/ICTCM/i/20/S118.html] Abstract: This paper is a conceptual articulation and exploration of what technological pedagogical content knowledge (TPCK) would look like in statistics education. Connections to the GAISE College Report (ASA, 2005) and other literature are synthesized and various concrete examples are provided and discussed.
Lesser, L. (2009). Equity, Social Justice, and the Mission of TODOS: Connections and Motivations. Teaching for Excellence and Equity in Mathematics, 1(1), 22-27. Abstract: Equity and social justice are shown to be intertwined with each other and with the TODOS mission. Also, Shaughnessy (2007) and the author’s pilot survey of in-service secondary teachers suggest interaction (or even interference) between students’ prior concepts of fairness and certain mathematics/statistics topics. Recommendations for exploration are provided (as well as pre-reading and post-reading discussion questions).
Lesser, L. (2009). Social Justice, Gender Equity, and Service Learning in Statistics Education: Lessons Learned from the DOE-Funded Project ACE (ACtion for Equity). Proceedings of the 2008 Joint Statistical Meetings, Section on Statistical Education, pp. 424-431. Alexandria, VA: American Statistical Association. [ISBN: 978-0-9791747-5-9]
Abstract: Equity, service learning, and social justice are powerful vehicles for motivating students to take statistics seriously and also for empowering citizens with the statistical literacy needed to be able to speak out more intelligently against injustices they may uncover. This paper begins by outlining key references from the recent, rapidly emerging literature on these topics in statistics education. A major grant on the author’s campus dealing with gender equity in the context of STEM (science, technology, engineering, and mathematics) fields yielded a natural vehicle to implement many aspects of these themes in a redesigned introductory statistics course for pre-service elementary and middle school teachers. After describing the grant, the student population, and features of the course, this paper presents the results of a quantitative pretest-posttest survey as well as representative narrative data such as artifacts (e.g., student reflection papers) and a peer observation. Further connections and reflections are made and situated in the literature.
Garfunkel, Solomon; Malkevitch, Joseph; Lesser, Lawrence M.; Moore, David S.; Taylor, Alan D.; Conrad, Bruce P.; Brams, Steven J.; Gallian, Joseph; Campbell, Paul J.; (2009). For All Practical Purposes (8th edition of critically-acclaimed, top-selling math-for-liberal-arts textbook). New York: W. H. Freeman and Company (with COMAP: Consortium for Mathematics and its Applications). [For this major revision of the 23-chapter book, I had sole responsibility for the 4 statistics chapters, pp. 147-282, SC-1, A8-A11. One of those chapters (Ch. 5) was selected as the featured FAPP chapter for the company’s webportal (expected to launch spring 2009 at http://www.whfreeman.com/newcatalog.aspx?technology=Portal.]
Abstract: The four statistics chapters cover distributions (including graphical and numerical summaries of quantitative data), correlation, regression, sampling, experiments, observational studies, confidence intervals, and probability. I made extensive refinements throughout all four of these chapters, and roughly 30% of the examples, exercises, spotlights, etc. have been changed or replaced. I added new technology spotlights (covering graphing, scientific, and nonscientific calculators) to aid in calculating standard deviation, five number summary, correlation, line of best fit, and combinatorics. Standard deviation and correlation formulas are now provided in both computational as well as conceptual forms. Coverage of sample space, probability rules, combinatorics, and descriptive statistics was expanded. Connections to history, multiple representations, etymology, culture, and the classroom have been added to make these chapters more engaging for readers and more friendly for English language learners.
Lesser, Lawrence & Pearl, Dennis (2008). Functional Fun in Statistics Teaching: Resources, Research, and Recommendations. Journal of Statistics Education, 16(3), http://www.amstat.org/publications/jse/v16n3/lesser.pdf (or http://www.amstat.org/publications/jse/v16n3/lesser.html).
Abstract: This paper presents an overview of modalities that can be used to make learning statistics fun. Representative examples or points of departure in the literature are provided for no less than 20 modalities. Empirical evidence of effectiveness specific to statistics education is starting to emerge for some of these modalities – namely, humor, song, and cartoons. To reinforce their effectiveness as an intentional teaching tool, the authors offer practical implementation tips.
Kosheleva, Olga; Lesser, Lawrence; Munter, Judith; Trillo, Sylvia. (2008) Parent Power Nights: A Vehicle for Engaging Adults/Families in Learning Mathematics. Adults Learning Mathematics International Journal, 3(2b), 36-52.
Abstract: Located on the U.S./México border, The University of Texas at El Paso offers academic programs in K-12 school teacher preparation. Many of the courses integrate parents and families into teacher preparation courses. One example of effective adult/community learning is the “Parent Power Night” (PPN) component. This model builds a learning community, engaging university faculty members with pre-service teachers and family members in effective teaching/learning activities. Pre-service teachers are concurrently enrolled in mathematics content and pedagogy courses, taught together in a “block” on the campus of a public school. PPN activities aim to engage parents and community members together with the university students in meaningful investigations of mathematical concepts. Preliminary evidence (e.g., from forming categories out of the responses from semi-structured interviews and surveys of individuals and small groups at different times) suggests that PPN activities have impacted knowledge and attitudes towards mathematics of participating parents, children and pre-service teachers from this predominantly Hispanic, high-poverty area. An unanticipated outcome has been the impact on adults with limited previous formal education; many acquired the knowledge necessary to understand rather sophisticated mathematics concepts their children were learning in school. The paper will discuss instructional methods used and implications for effective adult/family learning of mathematics content in Hispanic communities.
Tchoshanov, Mourat; Lesser, Lawrence; Salazar, James (2008). Teacher Knowledge and Student Achievement: Revealing Patterns. Journal of Mathematics Education Leadership, 10(2), 39-49.
Abstract: University researchers and teacher facilitators implemented a state-funded professional development project during the 2005-06 academic year to help county middle school teachers improve student achievement in mathematics. In this paper, we discuss lessons and results from this innovative model, whose iterative cycle includes teacher content knowledge, item analysis from a high-stakes test, pedagogical content knowledge, big mathematical ideas behind test items, and designing/ implementing/ reflecting on lessons to address critical problem areas in student learning and understanding.
Lesser, Lawrence (fall 2008). Equity, Social Justice, and the Mission of TODOS. Noticias de TODOS: News from TODOS Mathematics for All, 4(2), 7-9.
Abstract: While social justice may be perceived as a more “radical” or marginal realm than equity, these realms are shown to be intertwined with each other and with the mission of TODOS. Furthermore, the author’s exploratory pilot survey of a class of (N = 8) inservice secondary teachers reinforces evidence from the K-12 research literature (e.g., Shaughnessy 2007) that people’s (prior) concepts of fairness may impact how they encounter standard mathematics concepts, thus providing another reason to take concepts of social justice seriously. Resources are provided for those interested in beginning to learn about social justice teaching in mathematics/statistics education.
Lesser, Lawrence and Melgoza, Lorraine (2007). Simple Numbers: ANOVA Example of Facilitating Student Learning in Statistics. Teaching Statistics, 29(3), 102-105.
Abstract: An improved pedagogical sequence of datasets was created to increase secondary school inservice teachers’ conceptual intuition for one-way analysis of variance(ANOVA). During one 80-minute meeting of a course on statistical methods in mathematics education research, the teachers (n = 12) were given a pre-survey to assess their intuition about concepts of ANOVA (e.g., “between group variation” versus “within group variation”), then the intervention (individually answering questions about the structured sequence of datasets), then a post-survey. The intuition gain was about one point (on a 7-point Likert scale), but because of the small class size, the one-tailed paired t-test value (t = 1.363, df = 10) did not reach statistical significance (p = .101). There was, however, a statistically significant result (p = .0012) that teachers felt it was helpful that the numbers in the datasets were ‘simple’ (e.g., integer means and standard deviations).
Lesser, Lawrence (fall 2007). Using ‘Objects’ to Object to Objectification. Teaching Tolerance. No. 32, p.15.
Abstract: An activity tested and easily tailored for multiple grade levels uses the vehicle of classifying a variety of functions (or even numbers) by a variety of traits as a way to deepen understanding about both mathematics and tolerance.
Lesser, Lawrence (2007). Learning Stats is Fun … with the Right Mode. Stats, 48, pp.7-11,
Lesser, Lawrence (2008). Even More Fun Learning Stats. Stats, 49, pp. 5-8, 19, 27.
Abstract: Definitive comprehensive overview of modalities that can be use to making learning statistics fun, including humor, song, books, games, game shows, literature, word games, movies, videos, food, and celebrations. Most of the strategies are research-based and/or classroom tested and the paper includes a lengthy annotated bibliography.
Lesser, Lawrence (2007). Critical Values and Transforming Data: Teaching Statistics with Social Justice. Journal of Statistics Education, 15(1), 1-21. www.amstat.org/publications/jse/v15n1/lesser.pdf.
Abstract: Despite the dearth of literature specifically on teaching statistics for social justice, there is precedent in the more general realm of teaching for social justice, or even teaching mathematics for social justice. This article offers an overview of content examples, resources, and references that can be used in the specific area of statistics education. Philosophical and pedagogical background resources are given, definitional issues are discussed, and potential implementation challenges are addressed. A substantial bibliography of print and electronic resources is provided.
Lesser, Lawrence (2007). Using Graphing Calculators to do Statistics: A Pair of Problematic Pitfalls. Mathematics Teacher, 100(5), 375-378.
Abstract: We explore and discuss pedagogical opportunities presented by two subtle graphing calculator pitfalls that can be readily encountered in the secondary school classroom when doing statistics on common (TI) calculators: (1) confusion about bounds when computing cumulative probabilities for the normal distribution, and (2) confusion about the order of variables when computing regression lines of best fit to a dataset.
Lesser, Lawrence and Winsor, Matthew (2006). Interactive Representations of the Big Six Trigonometry Functions: Connections to Geometry and Language. ON-Math: Online Journal of School Mathematics, 5(1).
Abstract: Trigonometry classes can explore interactive sketches which allow them to connect the secant and tangent trigonometry functions to those words in a geometry context, and connect all six basic trigonometry functions (sin, cos, tan, cot, sec, csc) to specific segment lengths in a single simple diagram. The interactive nature of the diagram will also allow students to make connections to major inequalities and identities. The paper concludes with discussion and another applet using the applied context of the Ferris Wheel Problem.
Lesser, Lawrence and Tchoshanov, Mourat (2006). Selecting Representations. Texas Mathematics Teacher, 53(2), 20-26.
Abstract: This paper is an accessible overview of key research (by the authors and others) and pedagogical considerations related to choosing representations and representational sequences in school mathematics. Examples are explored from a variety of content areas.
Lesser, Lawrence and Blake, Sally (2006). Mathematical Power: Exploring Critical Pedagogy in Mathematics and Statistics. In C. Rossatto, R.L. Allen, M. Pruyn (Eds.), Re-inventing Critical Pedagogy: Widening the Circle of Anti-Oppression Education, pp. 159-173. Lanham, MD: Rowman & Littlefield.
Abstract: We discuss how negative attitudes are perpetuated that many students have about mathematics and their mathematical abilities. Informed by concrete classroom experiences, we then discuss how the tools of mathematics and mathematical reasoning can be applied towards culturally-relevant pedagogy and teaching for social justice to confront this and help students utilize the opportunities for empowerment and success they deserve in mathematics class and in life.
Lesser, Lawrence (2006). Book of Numbers: Exploring Jewish Mathematics and Culture at a Jewish High School. Journal of Mathematics and Culture, 1(1), 8-31. [the first juried comprehensive article on Jewish culturally relevant mathematics]
Abstract: At a pluralistic Jewish community high school in the southern US, the author sought, adapted, and integrated into his teaching examples of culturally relevant mathematics (in ways adaptable for other grades or cultures). Topics/techniques explored included: quotations about mathematics from traditional Jewish sources and sages, mathematical “firsts” (first statistical graphic, first fair division problem, etc.), counting (permutations, marking time, etc.), connecting mathematical and Jewish ideas about the infinite and about pi, mathematical modeling (e.g., Mikva’os 7:2), use of geometry in Judaism, connections between structures of logic used in mathematics and Judaism, and connections to Jewish text, customs or games (e.g., dreidl). In addition to their intrinsic interest and value, these enhancements connected to school culture/activities and appeared to help motivate additional students towards a broader view of and deeper engagement with mathematics, and possibly with Judaism as well. This article offers both scholarly background as well as a collection of diverse classroom-tested examples.
Lesser, Lawrence (2006). Engaging the Intuition in Statistics to Motivate. Juried paper (also presented at 50th national AP Conference) published on AP Statistics Course Home Page at College Board AP Central.
Abstract: An overview of how to motivate and bring intuition to concepts that are initially nonintuitive or even counterintuitive to students. Examples are provided that use a variety of means, including using multiple representations, intuitive analogies, and using(and resolving) counterintuitive examples. A thorough bibliography of additional resources and references is included.
Lesser, Lawrence (with Liz Rayas, Gabriel Trujillo, Aracely Vargas, Yogesh Velankar) (2006). Teachers’ Technology Class Continues Discussion of Pitfalls. Mathematics Teacher, 99(5), 340-342.
Abstract: The author explored pitfalls of technologies common in the secondary classroom with his “technology in the math classroom” class for preservice and inservice secondary teachers. Examples involving the TI 83/84 graphing calculator include regression syntax, nonzero value for sin(4*pi), a defined derivative at an absolute value function’s corner, graphical display of discontinuous functions, and order of operations. Other technologies for which pitfalls were identified include Excel, Mathematica, and even presentation/projection technology. Discussion is augmented with contributions by inservice high school teachers.
Lesser, Lawrence and Tchoshanov, Mourat (2005). The Effect of Representation and Representational Sequence on Students’ Understanding, 7-page research report. In G. M. Lloyd, M.R. Wilson, J.L.M.Wilkins, & S.L. Behm (Eds.), Proceedings of the 27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. [proceedings at http://convention2.allacademic.com/index.php?cmd=pmena_guest and on CD-ROM].
Abstract: This study investigates the effect of representational sequence on students’ understanding of mathematical concepts. Pilot studies were conducted with 129 high school students on solving inverse trigonometric identities and with 10 pre-service secondary teachers on representing Simpson’s Paradox. Structured activities with a variety of representations and representational sequences were used to examine the impact on students’ learning. This study also includes outcomes of surveys of 8 middle school teachers on different aspects of using representations in mathematics classroom. Our ongoing work finds this impact significant and claims that particular representational sequences need to be sensitive to specific content, learning outcomes, student prior knowledge and learning style.
Abstract: The author discusses several specific ways in which he has attempted to bridge theory and practice in teaching courses for preservice elementary teachers and courses for preservice secondary teachers. The column also references pitfalls and suggestions from the literature on this topic.
Lesser, Lawrence (March 2005). Mathematical Knowledge for Teaching, Theory and Practice” column in [Association of Mathematics Teacher Educators] Connections, 14(2), pp. 8-9. Also at: http://amte.sdsu.edu/resources/Mar05.pdf.
Abstract: the example of Simpson’s Paradox is used as a vehicle to discuss the many levels and facets of specialized mathematical/statistical knowledge needed for teaching, beyond just general mathematical/statistical maturity.
Lesser, Lawrence (Spring 2005). Illumination Through Representation: An Exploration Across the Grades, Statistics Teacher Network, 66, pp. 3-5.
Abstract: To support the newest process standard of NCTM (National Council of Teachers of Mathematics), the potential of multiple representations for teaching repertoire is explored through a real-world phenomenon for which full understanding is elusive using only the most common representation (a table of numbers). The phenomenon of "reversal of a comparison when data are grouped" can be explored in many ways, each with their own insights, including: table, platform scale, trapezoidal representation, unit square model, probability (balls in urns), and verbal form. Lesser also commented on this topic in a letter published in The American Statistician (November 2004, p. 362).
Lesser, Lawrence (Winter 2004). Take a Chance by Exploring the Statistics in Lotteries, Statistics Teacher Network, 65, pp. 6-7.
Abstract: This article gives intuition for the magnitude of the MegaMillions jackpot probability and then goes on to show how a lottery can be used to explore all the major topics of an introductory statistics course.
Lesser, Lawrence M. and Nordenhaug, Erik. (November 2004). Ethical Statistics and Statistical Ethics: Making an Interdisciplinary Module. Journal of Statistics Education. (14,000+ word article published by the American Statistical Association on the world-wide web at: http://www.amstat.org/publications/jse/v12n3/lesser.html)
Abstract: Describes an innovative curriculum module the first author created on the two-way exchange between statistics and applied ethics. The module, having no particular mathematical prerequisites beyond high school algebra, is part of an undergraduate interdisciplinary ethics course which begins with a 3-week introduction to basic applied ethics taught by a philosophy professor (the second author), and continues with 3-week modules from various other professors. The first author’s module’s emphasis on conceptual and critical thinking makes it easily adaptable to service-level courses as well as readily expandable for more mathematically sophisticated audiences. Through in-class explorations and discussions, the module made connections to contemporary topics such as the death penalty, equal pay for equal work, and profiling. This article shares resources, strategies and lessons learned for instructors wishing to develop their own specific modules of various lengths, but also contains valuable, provocative material and framework ideas for all teachers and practitioners of statistics.
Lesser, Larry. (Fall 2004). Slices of Pi: Rounding Up Ideas for
Celebrating Pi Day. Texas
Mathematics Teacher, 51(2),
6-11. (the issue is also available at http://www.tenet.edu/tctm/downloads/TMT_Fall_04.pdf
) Abstract: A
creative, comprehensive user-friendly overview of
ideas, activities and resources for educators (particularly secondary school
teachers) to implement “Pi Day” (3/14) celebrations at their schools. Connections are made to many realms, including: literature,
music, art, food, humor, contests, mnemonics, mathematics history, media,
hands-on activities, etc.
Lesser, Larry. (Fall 2003). A Whole Lotto Education! Texas Mathematics Teacher, 50 (2), pp. 12-15). (the issue is also available at http://www.tenet.edu/tctm/downloads/journal_fall03.pdf ) Abstract: Describes classroom explorations of the interpretation and calculation of probabilities involved in a representative state lottery. TI-83 calculator commands are given for simulating drawings as well as for calculating relevant probabilities using the binomial, geometric, Poisson, and other distributions.
Lesser, Lawrence M. (2002). Letter to the Editor. [critique/response to Sowey (2001) “Striking Demonstrations in Teaching Statistics” , JSE, 9(1)]. Journal of Statistics Education, 10 (1). Abstract: Frameworks for "striking examples" and "counterintuitive examples" are further articulated in light of recent work of E.R. Sowey, and additional examples are contributed. The importance of classifications by Lesser (1994) is reinforced, and there is a review of effort that has been made to identify and use such demonstrations in teaching.
Lesser, Lawrence M. (Winter 2002). Stat Song Sing-Along! STATS, #33, pp. 16-17. Abstract: Examples of highly creative lyrics (e.g., educating “The Gambler” about playing the lottery) are given that are rich in statistical content and/or related to current events.
Lesser, Lawrence M. (2001). Representations of Reversal: An Exploration of Simpson's Paradox. In A. A. Cuoco and F. R. Curcio (Eds.), The Roles of Representation in School Mathematics, pp. 129-145 [chapter of the NCTM's juried annual yearbook]. Abstract: To support NCTM's newest process standard, the potential of multiple representations for teaching repertoire is explored through a real-world phenomenon for which full understanding is elusive using only the most common representation (a table of numbers). The phenomenon of "reversal of a comparison when data are grouped" is explored in surprisingly many ways, each with their own insights: table, circle graph, slope & correlation coefficients, platform scale, trapezoidal representation, unit square model, probability (balls in urns), matrix determinants, linear transformations, vector addition, and verbal form. For such a mathematically-rich phenomenon, the number of distinct representations may be too large to expect a teacher to have time to use all of them. Therefore, it is necessary to learn which representations might be more effective than others, and then form a sequence from those selected. Pilot studies were done with pre-service secondary teachers (n1 = 7 at a public research university and n2 = 3 at a public comprehensive university) on exploring a sequence of 7 different representations of Simpson’s Paradox. Students tended to want to stay with the most concrete and visual representations (note: a concrete-visual-analytic progression may not be expected to apply in the usual manner in the particular case of Simpson’s Paradox).
Lesser, Lawrence M. (Autumn
2001). Musical Means:
Using Songs in Teaching Statistics. Teaching Statistics, 23 (3),
Abstract: Students’ ready understanding of and interest in the context of songs and music can be utilized to motivate all grade levels to learn probability and statistics. Content areas include generating descriptive statistics, conducting hypothesis tests, analyzing song lyrics for specific terms as well as “big picture” themes, exploring music as a data analysis tool, and exploring probability as a compositional tool. Musical examples span several genres, time periods, countries and cultures. [note: this appears to be the first refereed comprehensive article on using song in the statistics classroom]
Lesser, Lawrence M. (May 2000). Sum
of Songs: Making Mathematics Less Monotone! Mathematics Teacher,
Abstract: Mathematics students and teachers with even minimal musicianship can enjoy mathematical connections and motivations involving existing popular songs, raps or new words for existing songs. This article provides strategies, activities and examples as well as resources to "do it yourself." The article offers song-based problem solving, critical thinking and enrichment activities, and includes several highly original math lyrics (such as "American Pi", which can be sung to the tune of the song "American Pie" -- a #1 hit for Don McLean in 1972 and a Top-30 hit for Madonna in 2000) to support the multiple intelligences-based learning of mathematics procedures, content, process, and history. [note: this appears to be the first refereed comprehensive article on using song in the mathematics classroom]
Lesser, Lawrence M. (January 2000). Reunion
of Broken Parts: Experiencing Diversity in Algebra. Mathematics
Teacher, 93(1), 62-67.
Abstract: Algebra offers opportunities for all students to engage the richness of diversity without needing extra class time. Examples are illustrated from multiculturalism/history (e.g., solving linear equations using Egyptian method of "false position"), multiple representations (e.g., geometric representation of completing the square), and the object concept of functions (e.g., classifying a function by a given property).
Lesser, Lawrence M. (December 1999). Making
the Black Box Transparent. Mathematics Teacher, 92(9), 780-784.
Abstract: Line of best fit, interpolating polynomials, and complete graphs provide fresh opportunities for viewing technology and mathematical theory as partners rather than as competitors. In particular, when the computer outputs a line of best fit, a student may engage the formulas involved using algebra instead of calculus (which nicely complements the Summer 1999 Teaching Statistics article). When the computer crunches an interpolating polynomial, a student may do the same using the intuitive Lagrange pattern of factored form. And finally, a student can more effectively utilize a graphing calculator to graph functions such as polynomials by applying a theoretical result (accessibly provable using the Factor Theorem and the triangle inequality) to ensure the entire function is within the rectangular viewing area.
Lesser, Larry (Summer
1999). The Y’s and Why Not’s of
Line of Best Fit. Teaching Statistics, 21(2),
Abstract: This article presents a sequence of explorations and responses to student questions (Why not use perpendicular deviations? Why not minimize the sum of the vertical deviations? Why not minimize the sum of the absolute deviations? Why minimize the sum of the squared deviations?) about the rationale for the commonly used tool of line of best fit. A noncalculus-based motivation is more feasible than is often assumed for each aspect of the least-squares criterion “minimize the sum of the squares of the vertical deviations between the fitted line and the observed data points.”
Lesser, Lawrence M. (May 1999). Exploring
the Birthday Problem with Spreadsheets. Mathematics Teacher, 92(5),
Abstract: The Birthday Problem is “How many people must be in a room before the probability that some share a birthday (ignoring the year and ignoring leap days) becomes at least 50%?” Multiple approaches to the problem are explored and compared, addressing probability concepts, problem solving, modelling assumptions, approximations (supported by Taylor series), recursion, (Excel) spreadsheets, simulation, and student preconceptions. The traditional product representation that yields the exact answer is not only tedious with a regular calculator, but did not provide insight on why the answer (23) is so much smaller than most students' predictions (typically, half of 365). A more intuitive (but slightly inexact) approach synthesized by the author focuses on the total number of "opportunities" for matched birthdays (e.g., the new "opportunities" for a match added by the kth person who enters are those that the kth person has with each of the k-1 people already there). The author followed the model of Shaughnessy (1977) in having students give predictions in advance of the exploration and these written data (as well as interview data) collected from students indicated representative multiplier or representative quotient effects, consistent with the literature on misconceptions and heuristics. Data collected from students after the traditional and “opportunities” explorations indicate that a majority of students preferred the opportunities approach, favoring the large gain in intuition over the slight loss in precision.
Lesser, Lawrence (1999). Investigating the Role of Standards-Based Education in a Pre-Service Secondary Math Methods Course. In Myra L. Powers and Nancy K. Hartley (Eds.), Promoting Excellence in Teacher Preparation: Undergraduate Reforms in Mathematics and Science [juried monograph for NSF-funded Rocky Mountain Teacher Education Collaborative, also ERIC ED439089], pp. 53-64. Ft. Collins, CO: Colorado State University. Abstract: A case study was conducted on a math methods class for preservice secondary teachers that were exploring ideas and implementation of standards-based education. A variety of qualitative data was collected and analyzed about students’ experiences with performance-based assessment scoring rubrics as well as with state and national mathematics content standards, in a context of evolving professional identity and commitment.
Mayes, Robert L. and Lesser, Lawrence M. (1998). ACT in Algebra: Applications, Concepts, and Technology in Learning Algebra. McGraw-Hill. Abstract: Progressive college algebra textbook progressive in its incorporation of technology, having mathematics introduced by applications rather than by definitions, conceptual connections, etymology, math history, etc. The book has a realistic treatment of the place of factoring, having a chapter on factoring-dependent mathematics for those students who need that material for later mathematics courses, but a chapter that can be omitted without loss of continuity for more applied or terminal versions of this course.
Lesser, Larry (Spring
1998). Countering Indifference
Using Counterintuitive Examples. Teaching Statistics, 20(1),
Abstract: This article explains and synthesizes two theoretical perspectives on the use of counterintuitive examples in statistics courses, using Simpson’s Paradox as an example. While more research is encouraged, there is some reason to believe that selective use of such examples supports the constructivist pedagogy being called for in educational reform. A survey of college students beginning an introductory (non-calculus based) statistics course showed a highly significant positive correlation (r = .666, n = 97, p < .001) between interest in and surprise from a 5-point Likert scale survey of twenty true statistical statements in lay language, a result which suggests that such scenarios may motivate more than they demoralize, and an empirical extension of the model from the author’s developmental dissertation research. [this paper was subsequently selected by the editors for inclusion in Getting the Best from Teaching Statistics, a collection of the best articles from volumes 15-21] available at: http://www.rsscse.org.uk/ts/gtb/lesser.pdf
Lesser, Lawrence M. (February 1998). Technology-Rich
Standards-Based Statistics: Improving Introductory Statistics at the
College Level. Technological Horizons in Education Journal, 25(7),
Abstract: A university’s introductory statistics course was redesigned to incorporate technology (including a website) and to implement a standards-based approach that would parallel the recent standards-based education mandate for the state’s K-12 schools. The author collected some attitude (pre and post) and performance (post only) data from the “treatment” section and two “comparison (i.e., more traditional)” sections. There was a pattern of positive attitude towards the redesigned aspects of the course, including group work, lab and project emphasis, criterion-referenced assessment and examples from real-life. On the three problems given to the three sections at the end of the course, the only significant ANOVA (F2, 101 = 4.2, p = .0168) involved the treatment section scoring higher than the other sections. This occurred on a problem involving critical thinking (with a graphic from USA Today), an emphasis supported by the particular standards of the redesigned course.
Lesser, Lawrence (Nov. 1997). Exploring
Lotteries with Excel. Spreadsheet User, 4(2), 4-7.
Abstract: Spreadsheets are used to explore the lottery, addressing common misconceptions about various lottery "strategies" and probabilities and providing real-world applications of topics such as discrete probability distributions, combinatorics, sampling, simulation and expected value. Additional pedagogical issues are also discussed. Examples discussed include the probability that an integer appearing in consecutive drawings, the probability that a single 6-ball drawing includes at least two consecutive integers, the probability that exactly one person wins the jackpot, and the probability that a frequent player eventually wins the jackpot.