Be Wary of H.S. Statistics

By Jerome Dancis, Associate Professor Emeritus, Math Dept., Univ. of MD

Reading and Arithmetic-level data analysis (Statistics) is very important for all. This means knowledge and understanding of averages, medians, percentiles, box and whisker diagrams; also being able to read and draw a variety of graphs, charts and tables[1] as well as proficiency with percents and decimals and the big bugaboo, word problems. This is more important than Algebra. The rush to Algebra, should be replaced by the careful development of student proficiency in reading Arithmetic word problems and Arithmetic-level data analysis.

Unfortunately many high school and some college graduates are not well versed in Arithmetic data analysis. As a U.S. Dept. of Education study[2] noted: "… far fewer [Americans] are leaving higher education with the skills needed to comprehend routine data, such as reading a table about the relationship between blood pressure and physical activity, … 'What's disturbing is that the assessment is ... [designed] to test your ability to read labels,' [Mark S. Schneider, commissioner of education statistics] added." [3]

The Common Core Math curriculum (which is about to become the next national [not federal] math curriculum) does not include students knowing percentiles or knowing that a 50% off sale means half price or even that 50% = 1/2.

The Common Core’s Glossary’s definition[4] of “first quartile” is WRONG!

A prerequisite for understanding random variables in Statistics, is understanding (the far simpler) variables (the x’s) in Algebra. Proficiency in translating word problems into Algebraic formulas is the basis for writing formulas for spreadsheets.

Colleges’ attitude to freshmen, with zero K-12 Statistics, is: No Statistics; no problem. Colleges are reasonably successful at teaching Statistics – at least to those students, who are fluent in Arithmetic and Algebra. Many are not [5].

The choice, of which statistical method to use usually depends on context, context, context, that is, on how the information will be used. For example, on my campus, psychology majors are required to take PSYC 200, “Statistical Methods in Psychology”, which builds on PSYC 100, “Introduction to Psychology”. Business majors are required to take BMGT 230, “Business Statistics”. Sociology majors take SOCY 201, “Introduction Statistics for Sociology”, which builds on SOCY 100. This spring, we taught 41 classes of these three specialized beginning courses in Statistics. In addition the Mathematics Dept. taught 8 classes of STAT 100. “Elementary Statistics and Probability”.

An eleventh grader, doing well in Algebra II, has several math options for Grade 12, including Pre-Calculus and AP Statistics. Learning Pre-Calculus will make him/her fully ready for all college majors, including STEM majors. But, taking AP Statistics will likely put him/her at-risk for college majors in statistics and engineering.

Common Core Math Standards prescribed a Statistics strand for high and middle school. Also, a weak Algebra II syllabus. Time on Statistics will take time away from Algebra. This may increase the numbers of students needing to retake high school Algebra II in college.

That college math professors consider statistics and probability to be optional courses in high school is reflected in their statement that statistics and probability are two of “several mathematics courses that could be considered reasonable for study once students have achieved a strong background in algebra and geometry” [6].

That students are not obtaining competency with Arithmetic-level data analysis is indicated by the following Problem 1, which stymied more than 5 of 8 (65%) Grade 9 students, when it was field tested in Maryland (MD).

Problem 1.[7] "The table below shows how a typical household spends money on utilities.

Utility Percentage of Total Utility Costs

Lighting 6

Refrigeration 9

Water heating 14

Appliances 27

Heating and cooling 44.

A typical household spent $1,400 on utilities last year. If there are no significant changes in their utility usage for this year, how much should they budget for heating and cooling their home this year?

[Multiple Choice] F $196   G $378   H $616    J $784 "

Comment.  Students had calculators to calculate 44% of $1400 or they might simply notice that H $616  is the only choice that is a little less than $1400/2.

Data analysis is often too tricky for high school. It is even too tricky for the writers of the State of Maryland High School Assessment [MD HSA] on [Some concepts from] Functions, Algebra, Probability and Data Analysis. (Passing this assessment is a high school graduation requirement.) For example:

MD Assessment Item on Data Analysis [8]. “In a small town, 250 randomly sampled registered voters were asked to state whether they would vote “Yes” or “No” on Measure A in the next local election. The table below shows the results of the survey.

VOTER SURVEY RESULTS

Yes No Undecided

96 34 120

There are 5,500 people expected to vote in the next election. Based on the data, how many people will vote “No” on Measure A in the next election?”

Students who answered 2,112, were marked correct on the 2007 MD state assessment. To obtain this answer of 2,112, students are expected to make a number of unwarranted and usually incorrect assumptions [9]. But, students who answer 2,112, on a college political science exam will likely be marked wrong; a correct answer would be: not enough information is provided for the list of reasons noted in the footnote. [10]

Again, Data analysis is often too tricky for high school: UMCP Physics Professor, Tom Cohen's, observations of his child (a student in Montgomery County Public Schools, MD) doing her Algebra/ data analysis homework on "best fit" lines:

"However, the way data analysis is taught and tested troubles me. ... [11] The issues are subtle and algebra one students are not prepared to deal with them. Thus, the students are being miseducated in data analysis and statistics.”

“In my view this treatment is worse than useless, it is positively destructive. Students are told in essence to plug things in which they don't understand and then to trust the answers. This is diametrically opposed to the critical reasoning about data analysis that we need to instill in students.”

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[1] Professor of Biological Sciences, at Towson State University, MD, Virginia Anderson, Ed.D., reported needing to provide extensive training to her college biology students in the reading and drawing of tables, graphs and charts.

[2] Conducted by the National Center for Education Statistics.

[3] "Literacy of College Graduates Is on Decline Survey's Finding of a Drop in Reading Proficiency Is Inexplicable, Experts Say", Washington Post, December 25, 2005; A12

[4] “First quartile. For a data set with median M, the first quartile is the median of the data values less than M. ! This leads to the first quartile for the data set {1, 3, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7} being 3. WRONG, it is 7.

[5] From a Univ. of MD, College Park [UMCP] instructor: ... [on] a recent quiz I gave in STAT 100 [our college version of AP Stat] ... . Most all [students] obtained the [correct] equation y = .52 + .7x, but many (over half of my class) missed points because they did not correctly graph the line. ... the line they drew ... was not y = .52 + .7x, but some arbitrary line. ..., a [student] commented, "Well, I graphed it on my calculator, shouldn't I get credit for that?" When I mentioned that I needed the scale, y-intercept, slope, etc. correctly graphed ... , the reply was, ".... the calculator doesn't show the scale so I don't know how to do that."

Another question on the quiz ... given that the equation ... y = -36.9 + 5.07x, [find y when x = 20]. I had more than one student who forgot his/her calculator say they couldn't do this problem because they didn't have a calculator (i.e. couldn't do decimal multiplication and addition).

FYI ... the text assumes students know [how to graph a line] before taking STAT 100..

[6] This was noted in the “[College Professors’ Concerns on] Mathematical Preparedness of Incoming College Freshmen”, the only statement on college preparedness issued by The MD/DC/VA Section of the Mathematical Association of America (MAA). On the web at: http://sections.maa.org/mddcva/HS_students.php

[7] Sample MD High School Assessment on Algebra and Data Analysis, Item 48 [2000]

[8] This is 2007 Public Release Algebra/Data Analysis Item #38 at

http://mdk12.org/assessments/high_school/look_like/2007/algebra/ftri38.html

This is also Item #37 at

http://www.mdk12.org/assessments/high_school/look_like/2007/algebra/hsaAlgebra.pdf

[9] The number of people, who will actually vote in the next election, is exactly (not just approximately) equal to the number expected to vote.

None of the undecided people will make up their mind and choose to vote “No” after the survey. This is rarely a true statement.

All of the surveyed people, who answered, “undecided” were actually undecided. Nobody said “undecided” as a polite way to say “None of your business”.

[10] Incorrect assumptions listed in preceding footnote.

[11] “In particular, the use of linear regressions (done by a calculator) to fit lines is not appropriate for algebra one students, in my view. The students are NOT taught what a "best fit" line means mathematically, how to judge whether the model fits the data well ... nor even given any clear way to understand whether the data ought to fit a line. If you ask the calculator for a line which will fit points which lie on a parabola the calculator will spit back a [misleading] line and the students will dutifully write it down.”