Comments by Dr. Jerome Dancis (associate professor of mathematics, UMCP)
Presented at the MD State Board of Education meeting, April 20, 2004
The level of the Math required of students by the MD Pretend Algebra and Pretentious Data Analysis exam is not higher than that required by the MD Functional Math exam.
It is the high reading level of the MD Pretend Algebra and Pretentious Data Analysis exam, which makes this exam more difficult than the MD Functional Math exam. If one looks at just the level of the math done by the students, then the MD Pretend Algebra Exam is not a step up. It is a step to the side. The exam sidesteps the math; this includes, but is not limited to leaving the bulk of the math to the hand calculators. Also the exam includes misleading and wrong Algebra and Data Analysis, which is counterproductive.
This is exemplified by the two math items showcased in MSDE's own pamphlet "Selected test items from the High School Assessments". Details below.
Showcase Algebra/Data Analysis Item #1 as written looks very impressive. (See it on the web at http://www.mdk12.org/mspp/high_school/look_like/2001/algebra/v39.html) But, after the camouflage is removed, the item is merely asking the student to count how many of the ten numbers listed, consist of the digits 1 to 8, only, (No 0 or 9). (Details in appendix.) This is why I use the phrase Pretentious Data Analysis to describe the so-called Probability and Data Analysis part of the exam.
The Data Analysis methodology is wrong-headed. Item #1 has Meghan run a simulation based on a survey, in order to find a probability. But, when survey results are known, the probability can and should be calculated directly, without a simulation. By sidestepping real probability and running a simulation instead, MSDE is requiring the training of impressionable students in wrong-headed Probability and Data Analysis
The real probability is .32768, (not .3 as required) it can be calculated using Grade 6 level probability. (Details in appendix.) But the correct probability of .32768 would be marked wrong by MSDE.
Showcase Algebra/Data Analysis
Item #2 .
(See it on the web at http://www.mdk12.org/mspp/high_school/look_like/algebra/v18.html)
In the showcase Response to Item #2, the graph was drawn by a graphing calculator; the student merely copied it. The student may obtain the correct answer of 4 days, by having the graphing calculator highlight the point, (4, 480) [where the lines cross in the graph]. The showcased response neglects to credit the graphing calculator.
Almost 3 out of 5 (58%) students omitted this Showcase Item #2, when it was field-tested in 2000. Almost half (47%) of the students, who answered this question, earned a score of 0 or 1 on a scale of 0 to 4, which "indicates little or no application of a reasonable strategy. [1] All this suggests that this item is considerably more difficult than most items. Most likely students can skip all items of this difficulty, and still pass the exam. All this suggests that Showcase Item #2 is not a typical item.
To its credit, the reading and writing level of this item is far higher than that anything on the MD Functional Math Exam. The beautifully written showcase Response to Item #2 earned full credit of 4 points. Only 4.2% of the students, who answered this question, earned this score of 4. This is only 2.2% of all the test-takers. This showcase Response to Item #2 is far from typical.
Doing Showcase Algebra/Data Analysis Item #2, by algebra, requires solving
160 + 80x = 120x. But solving 160 + 80x = 120x, or even 2x = 10, symbolically[2] is not on the exam's syllabus [3]. So students use graphing calculators as they side-step solving 160 + 80x = 120x symbolically. Some high schools Algebra I classes are skipping the symbolic solution and only teaching students the graphing calculator solution. This overuse of graphing calculators to solve: 160 + 80 = 120x, so as to sidestep the symbolic method, is a major reason that I use the phrase Pretend Algebra to describe the so-called Functions and Algebra part of the exam.
The method of choice, for solving the underlining problem of Showcase Item #2, should be arithmetic not Algebra, since the arithmetic solution is simpler and it provides more conceptual understanding of the underlining problem. (Details in appendix.)
A major purpose of Algebra is to provide simple solutions to difficult arithmetic problems; not to convert a simple arithmetic problem into a more involved Algebraic one, as is required by Showcase Item #2.
Showcase Item #2, to its credit, has the student "setting-up" the equations for a "word problem"; this is a crucial part of problem solving. It is a rare problem on the MD Algebra exams that requires this. As such, Showcase Item #2 is not representative of the MD Algebra tests. In Item #32, on the 2000 exam, the equations are provided.
A big No-No in real Algebra is never using the same variable to
mean two different things in the same
problem. But the showcase
Response to Item #2 has the student using the symbol "C" to represent the two different stores pricing
rules. This is also suggested by
MSDE's statement of showcase Item #2. Having the same variable mean two different things
occurs regularly in the statements of items on MD's Pretend Algebra exams. For example, it occurs in Item #17 in
the MD Pretend Algebra 2002 test and it occurred in Item #32 in the MD Pretend
Algebra 2000 test[4]. This type
of ambiguity should confuse students.
This suggests an item-writer, with little understanding of the very
basic, algebraic concept of "variables".
Bottomlines. The basic Pros and cons of the exams are:
MD's Functional Math Exam. PRO Students can do arithmetic by hand.
MD's Functional Math Exam. CON Students, who cannot read, can pass this exam.
MD Algebra exam PROs: 1. Students can use hand calculators to do simple math.
2. Students can read
3. Students know some background for real Algebra
4. Students know simple data analysis
MD Algebra exam CONs: 1. Some of the background for Algebra and much of the simple data analysis is misleading or wrong.
2. Students not required to do arithmetic by hand.
Bottomlines for taking a Real Algebra course.
MD's Functional Math Exam. PRO Students will have the arithmetic background needed for a real Algebra course. After completing a real Algebra course, they will have the Algebraic background needed for a non-trivial Data Analysis course.
MD's Functional Math Exam. CON Students may not have the background needed for non-trivial Algebraic word problems.
MD Algebra exam PROs: 1. Students have some facility with graphing calculators.
2. Students can read.
MD Algebra exam CONs: 1. Students not required to have the arithmetic background needed for a real Algebra course.
2. Students will need to unlearn the misleading and incorrect things required for the MD Algebra/Data Analysis exam. There are far worse errors than those listed in this report. See a few at "Beware the MD Algebra test " on my website http://www.math.umd.edu/~jnd
Appendix
Showcase Item #1. How to cut through the camouflage. Ignore the introductory paragraphs. Mis-translate the question line (at bottom) as "How many of the ten numbers listed, consist only of the digits coded for driving to work?" Go to the table for the code numbers for driving to work, they are the digits 1 to 8, only, (No 0 or 9). Count how many of the ten numbers listed, consist of the digits 1 to 8, only. It is 3 out of 10, or 3/10, the expected answer.
Field-testing of Showcase Item #1 occurred in 2001. Almost 1 out of 5 (19%) students omitted this item.
A somewhat better Showcase Item #1 would read: "Based on Meghan's simulation, what is an estimate of the probability that all 5 people in the group drive to work. I added the words in italics. The answer should be .3 ± .15. Probabilities based on simulation should come with error estimates.
A good Showcase Item #1 would be:
Problem (Real Probabilty). Given that 80% of workers drive to work, "what is the probability that all 5 people [workers] in a group drive to work?"
The probability, that a worker drives to work, is P(w) = 80% = 80/100 = 8/10.
How the individual workers, in a group get to work, are considered "independent events".
The formula, for the probability of two independent events occurring, is the product of the
probability of the individual events. In symbolic form:
Theorem. If A and B are independent events, then P(A, B) = P(A) x P(B).
This theorem is included in the California syllabus for Grade 6.
So the probability, that workers #1 and #2 (w1 and w2 ) both drive to work, is
P(w1, w2 ) = P(w1) x P(w2) = (8/10) x (8/10).
Proceeding in this manner, the probability, that all 5 workers in a group drive to work, is:
(8/10) x (8/10) x (8/10) x (8/10) x (8/10) = .32768.
Answer. The probability, that all 5 workers in a group drive to work, is .32768.
Showcase Item #2 (abridged to the underlining problem) . "Two bicycle shops build custom-made bicycles. Bicycle City charges $160 plus $80 for each day that it takes to build the bicycle. Bike Town charges $120 for each day that it takes to build the bicycle. … For what number of days will the charge be the same at each store?"
Arithmetic Calculations. For each day, Bike Town charges $120-$80 = $40 more than Bicycle City. But Bicycle City starts out higher with its extra $160 charge. It takes 160/40 = 4 days for the prices to equalize.
[1] Click
on "View rubric used to score this item on the web" at http://www.mdk12.org/mspp/high_school/look_like/algebra/v18.html
[2] Symbolic solution: 160 + 80x = 120x à 160 = 40x à 4 = x Not on syllabus for exam.
[3] As the
head of math instruction for the state, Donna Watts, put it: "[There is a]
limited population who can do math symbolically, the way mathematicians
do"
(http://www.washingtonpost.com/ac2/wp-dyn/A28031-2003Dec24?language=printer)
[4]
This occurred in the statement of Item #32 on
the sample MD Algebra 2000 test. I
informed MSDE of this error on Oct. 30, 2001. The following was added to the webpage for this item:
"The variable y is used to represent both the income for selling x tubes of toothpaste and the production cost for x tubes of toothpaste. This is an error in the use of a variable."