Lesser, Lawrence M. (2002). Letter
to the Editor. [ response (of nearly 2000 words) to Sowey
(2001) "Striking
Demonstrations in Teaching Statistics", JSE, 9(1) ]. Journal
of Statistics Education, 10 (1).
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. Examples of entertaining, highly original lyrics are given that are rich in statistical content and/or related to current events.
Lesser, Lawrence M. (2002). Ethical Statistics and Statistical Ethics: The Experience of Creating an Interdisciplinary Module. 2001 Proceedings of the American Statistical Association,Section on Statistical Education [CD-ROM]. Alexandria, VA: ASA.
Lesser, Lawrence M. (2001). Representations of
Reversal: Exploring Simpson's Paradox. In Albert
A. Cuoco and Frances R. Curcio (Eds.). The Roles of Representation in
School Mathematics, pp. 129-145 [the middle chapter of the NCTM's
juried annual yearbook].
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. 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. Connections to and implications
for research and teaching are discussed.
Lesser, Lawrence M. (Autumn 2001). Musical Means:
Using Songs in Teaching Statistics. Teaching Statistics, 23
(3), 81-85.
Students’ ready understanding of and interest in the
context of songs and music can be utilised to motivate all grade levels
to learn probability and
statistics. Content areas include generating descriptive
statistics, conducting hypothesis tests, analysing 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.
Lesser, Lawrence M. (May 2000). Sum of Songs:
Making Mathematics Less Monotone!
Mathematics Teacher, 93(5),
372-377.
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.
Lesser, Lawrence M. (January 2000). Reunion of
Broken Parts: Experiencing Diversity in Algebra. Mathematics
Teacher, 93(1), 62-67.
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.
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), 54-55.
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),
407-411.
The Birthday Problem (stated 60 years before) 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 does not provide insight on why the answer (23) is so much smaller
than most students' predictions (typically, half of 365). An
approximate, but more intuitive approach is then discussed that 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).
Lesser, Larry (Spring 1998). Countering Indifference
Using Counterintuitive Examples. Teaching Statistics,
20(1), 10-12.
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.
Lesser, Lawrence M. (February 1998). Technology-Rich
Standards-Based Statistics: Improving Introductory Statistics at
the College Level.
Technological Horizons in Education Journal,
25(7), 54-57.
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
(F(2,101) = 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 (November 1997). Exploring Lotteries
with Excel. Spreadsheet User, 4(2),4-7.
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.