Bewarethe MD Algebra test
AKA The Maryland High School Assessment on
Functions, Algebra, Data Analysis and Probability
By Jerome Dancis
There is almost no Algebra on the MD Algebra test.
But they fake it very well.
This is also a test of Data Analysis and Probability
Much fake or superficial.
Presented at the Nov., 2002 MD-DC-VA Sectional Meeting of the MAA
(The Mathematics Association of America --
The professional society for college mathematics education)
Dedicated to my mother, Anna Dancis
February 5, 1906 - October 22, 2002
Bewarethe MD Algebra test
By Jerome Dancis
Table of Contents
2. Algebra = X in One School, Y in Another (Excerpts from Washington Post)
wherein a MSDE official states that the MD Algebra test "doesn't have a lot of algebra, nor was it intended to".
3. The MD Algebra Test avoids Algebra conceptual understandings, and problem solving.
4. MD Algebra sample test Items #35 and #19 -- which are mathematically incorrect.
5. Data Analysis
6. Mis-education in data analysis
7. Bad Habits are Hard to Break
8. Purported emphasis on conceptual understanding.
9. Set the bar high in math, ensure students clear it by Chancellor emeritus Langenberg [of the University System of Maryland] (Baltimore Sun)
10. Purposes of the MD Algebra test
Appendix. When It Comes To Math, Words Count (Washington Post) , The math level on the sample MD algebra test is lower than the reading comprehension level.
1. Introduction. Comments by Dr. Jerome Dancis (associate professor of mathematics, UMCP) firstname.lastname@example.org Presented to the MD State Board of Education, Oct. 30, 2001
Re.: The Maryland High School Assessment on Functions, Algebra, Data Analysis and Probability test (MD Algebra test).
Last year, I served as the mathematics advisor for the California (CA) edition of Harcourt's new Grade 6 math book. This summer, I read all 49 questions, on the sample MD Algebra test. They reminded me of the ones in the Grade 6 math book. This book contains instruction for 30 of the 49 questions on the sample MD Algebra test.
Two weeks of additional instruction on jargon and how to read graphs would enable CA Grade 6 students to do 12 more simple problems, for a total of 42 out of 49. (For Question #25, students merely need to know that the impressive sounding-sentence "It is not continuous at x = 3." means simply that the graph [shown] has a jump or gap at x = 3.)
There is no real algebra (or real functions) on the sample exam; mainly, it is Grade 6 algebra preparation. The level of conceptual understandings and problem solving is very low, often trivial; certainly considerably lower than CA Grade 6.
The probability questions on the sample test are all "probabilities of simple events"-type, they require minimal conceptual understanding and problem solving ability. In contrast, the CA Grade 6 math book contains probabilities of compound events; this requires serious conceptual understanding.
The main knowledge needed for the state's test is reading, common sense and simple arithmetic (with reading including the reading of numbers off tables, charts and graphs). The reading level, demanded by this test, is higher than the algebra level. Calculation with symbols (x's) -- the core of Algebra -- is not included.
For students going on to college, the two main likely consequences of MD's Algebra test will be:
#1An increase in the already high numbers of students needing to take remedial math (Algebra) in college. Students will be able to score 100% on this "Algebra test" and still be placed in a real Algebra 1 course in college.
#2. Students wishing to pursue courses in science and engineering will arrive in college with even weaker backgrounds in algebra and trig. At UMCP, this is already sabotaging many a student's efforts to learn calculus, the gateway to engineering. It is getting out of hand. It will become even worse.
2.Excerpts from Washington Post front page Aug16, 2002
Algebra = X in One School, Y in Another
By Jay Mathews
After taking Maryland's state algebra test this year, Susan Gruenspecht's students at Westland Middle School in Montgomery County wanted to know: Where was the algebra?
They were not alone. University of Maryland mathematics associate professor Jerome Dancis called the test "pretend algebra." Montgomery County parent John Hoven, an economist, said most of the problems were what students in Singapore get in the fifth grade. Even one of the people responsible for the test, Maryland State Education Department official Gary Heath, said, "We would be the first to tell you it doesn't have a lot of algebra, nor was it intended to."
"A lot of algebra courses aren't really teaching algebra," said Tom Loveless, director of the Brown Center on Education Policy at the Brookings Institution in Washington. "And schools are giving up on kids who can't do basic arithmetic, putting them in phony algebra courses to cover up the problem."
But several experts note that most states either have no test to check what students have learned or have produced a test such as Maryland's, with just algebraic concepts.
"What happens all too often is you get some labels on courses and then they get backed down, so the kid might get an A but doesn't get an A's worth of algebra," said Jim Watts, vice president of the Atlanta-based Southern Regional Education Board, which works to improve schools in the region.
"The only way to make the standard true is to have a good end-of-course test," Watts said.
Maryland officials in the early 1990s found that math teachers in local school districts "were not supportive of a full-blown algebra test" that would be required for graduation, said Heath, chief of the arts and sciences branch of the State Department of Education.
Heath said that he has heard complaints about his state's test, officially known as the Algebra/Data Analysis Assessment, from a few parents and college professors, but that the State Board of Education agreed to start slow and introduce a standard algebra test once schools were ready for it.
In Virginia, an algebra test that would be required for high school graduation was approved in the 1990s by state school board members appointed by Govs. George Allen and James S. Gilmore III. The governors and their education advisers, like other southern leaders, said they could not persuade schools to upgrade their teaching without a strong state test.
Once Maryland's Algebra/Data Analysis Assessment becomes a requirement for graduation, probably beginning in 2007, officials will begin planning something more demanding, Heath said.
"Our long-term goal is to raise that bar, and the next level would be to have a full-blown algebra test," he said.
Teachers throughout the region say a strong test is needed to resist the temptation to reduce their courses to something closer to what the Maryland test measures.
"You get more kids in the class, and the next thing you know, the teacher gets pressure," said Vern Williams, a math teacher at Longfellow Middle School in Fairfax County. "They say, 'You guys have a different population now. You are going to have to change your teaching methods.' They will never use words like 'water down the course.' "
"Personally and professionally, I would prefer to have a more rigorous course," said Jean Bone, an algebra teacher and mathematics chairman at Westland Middle School. "But in some respects, I have no choice, because I have to follow the county curriculum."
3. The MD Algebra Test avoids Algebra conceptual understandings, and problem solving.
(An expansion of my comments presented to the MD State Board of Education on Oct. 30, 2001)
We will demonstrate this as we exam a problem from the MD HSA Algebra sample Test.
Nice Problem. A tube of tooth paste costs 90 cents to make, and sells for $2.50. The company has "fixed costs" (machinery or rent or whatever). of $3000. How many tubes of toothpaste does the company need to sell to cover/balance-out the fixed costs?
The profit on the sale of each tube is $2.50 - 0.90 = $1.60. Hence, the company will need to sell 3000/1.60 = 1875 tubes. (O.K. to use a calculator for the division only.)
This Nice Problem was not on the sample MD Algebra test; -- well not until all the conceptual understandings, and problem solving had been removed and after it had been rewritten in a long-winded and pretentious manner. ( I suggest that you read the first paragraph of the problem, then jump to the Pedagogical Analysis, below.):
Problem #32 on the sample MD Algebra test. (on the web at http://www.mdk12.org/mspp/high_school/look_like/algebra/v32.html):
"The income (y) for a particular toothpaste company is modeled by the equation y = 2.5 x dollars,
where y is the income for selling x tubes of toothpaste. The cost of producing toothpaste is
y = 0.9 x + 3000 dollars, where y is the cost of producing x tubes.
* How many tubes of toothpaste must be sold for the income to equal the production cost? Use mathematics to justify your answer. (If you solve the problem graphically, use the grid provided in the Answer Book to add to your written response.)
(Suggested graphing window 0 _ x _ 3000, 0 _ y _ 5000.)
* What is the income and production cost at the point when they are equal? (Underlines added)
* The company makes a profit when their income is greater than their production cost. What is the least number of toothpaste tubes the company can sell to make a profit? Use mathematics to justify your answer."
A PedagogicalAnalysis of Problem #32
A crucial part of problem solving is "setting-up" the equations for a "word problem". Also know as "modeling and interpreting real-world situations". This problem does not test this skill because the equations are provided. In sharp contrast, read the mis-claimed stated-expectation for this problem, on the state's website.:
Expectation 1.2: "The student will model and interpret real-world situations, using the language of mathematics and appropriate technology."
(Click, on "view Core Learning Goal, Expectation and Indicator this item tested" on right side at http://www.mdk12.org/mspp/high_school/look_like/algebra/v32.html )
In fact, I counted only one of the 49 problems on the sample MD Algebra test, which actually required the student to set up the equations.
Solving simple equations both by hand and with a graphing calculator, is an important part of real Algebra. Here the equation 2.5x = 0.9x + 3000 needs to be solved. But the students do not need to do the simple calculations; they are encouraged to use their graphing calculators (which provide graphs of the functions). In fact, I counted only two problems on the sample MD Algebra test, which required students to solve equations, none, which required students to solve equations without a graphing calculators.
Here, the thinking part was reduced to choosing the correct "window" to view on the graphing calculator. Even that was deemed too hard as suggested "window" ranges are supplied.
Economists use q for quantity and c for cost. Never the cryptic x for quantity and y for cost as in this problem. A needed skill, in setting up a problem, is to choose names of variables that assist in understanding the problem and the equations. But then graphing c = 0.9q + 3000 on a graphing calculator requires some conceptual understanding unlike y = 0.9x + 3000 which does not.
Another solution, which received the highest possible score when graded, is presented at http://www.mdk12.org/mspp/high_school/look_like/algebra/anchors/a32_score_3.html. Here the student typed the two given equations into the calculator and had the calculator list their table of values. The student then "scolled through the table until [the numbers for both Y's] were the same." Precious little [Grade 6] conceptual understanding and problem solving involved.
This avoidance of conceptual understandings, and problem solving is in sharp contrast to the Maryland State Dept. of Education statement:
"In all mathematics content standards, the emphasis is on achieving a balance among memorization of facts, proficiency with paper and pencil skills, appropriate use of technology, conceptual understandings, and problem solving" (Underline added). On the web at http://www.mdk12.org/mspp/standards/math/introduction.html.
A big No-No in real Algebra is never using the same variable to mean two different things in the same problem. This problem violates this rule, having y representing both "income" and "cost". This type of ambiguity often confuses students. This suggests a problem writer, with little understanding of the very basic algebraic concept of "variables" (the x's and y's) in algebra. Of course, problem writers, who actually understand Algebra would require more pay for each problem. This would reduce the profits of the profit-making, test-writing company.
The following was added to the webpage for this problem between June 2001 and March, 2002 (I informed them of this and all the errors listed above on Oct. 30, 2001):
"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."
4.MD Algebra sample test Item #35. "Look at the pattern below.
-1, 0, 3, 8, 15, ...
If the pattern continues, what will be the next term?
A 20, B 22 C 24, D 35"
The next term may be any number. All answers are correct answers. There are many patterns, all of which start with these five numbers, -1, 0, 3, 8, 15, and then have all sorts of next terms. All mathematicians know this. Many a Math Educator does not. This Item #35 is WRONG mathematics.
MD Algebra sample test Item #19.The table below shows the change in population for a group of mice over a 4-month period.
Population of mice
Month 0 1 2 3 4
Population 4 8 16 32 64
If this pattern continues, what will be the population of mice in Month 12?
The simplest pattern is that the number of mice doubles each month. But, there are many other patterns, all of which start with these five numbers, 4, 8, 16, 32, 64, and then have all sorts of next terms. All mathematicians know this. Many a Math Educator does not. This Item #19 is WRONG Mathematics and WRONG Biology.
Note that the biology is skipped, namely, the assumption that the growth rate is proportional to the population is not stated; it should have been stated.
Alternate pattern. Exactly half the population is female and each female gives birth to one female and one male mouse month. The original 4 mice were born in Month #0. But each mouse dies at age 6 months. Then the population for the first 5 months will be the same as in the table for Item # 19, but as the older mice die of old age, the population will no longer double each month.
This item is training students to avoid Biological reasoning and conceptual understanding. Counterproductive.
5.Data Analysis -- lines of best fit
MD Algebra sample test Item #17requires the student to "Write the equation for a line of best fit. Use mathematics to explain how you determined your line of best fit."
There are two answers on the state website which received the highest possible score of 3. In neither response does the student present a clue as to what the word "best" signifies or why a straight line fit might even be appropriate.
Best Response #1: The student graphs the points and simply chooses a line that looks good. The "official grader" for the website states:
" although the variables are not defined; this equation is the result of calculations using two points from the table that are near to the student's line of best fit. and the correct graph leads to an accurate y-intercept. The response demonstrates a complete understanding and analysis of the problem".
Best Response #2. The student enters the data points into the graphing calculator and the hits the "linear reg." button. The "official grader" for the website states: " although the student does not define the variables, . . The student then provides an undeveloped explanation for the context of the slope (" the slope represents that as the number of coins increases so does the weight") . The response demonstrates a complete understanding and analysis of the problem".
6. Mis-education in data analysis
This Best Response #2 is consistent with UMCP Physics Professor, Tom Cohen's, observations of his child (a student in Montgomery County) doing her "Algebra" homework on "best fit" lines, namely:
"However, the way data analysis is taught and tested troubles me. 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 (i.e. chi square or other statistical measure) 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 line and the students will dutifully write it down. 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.
Finally, significant time is spent on this activity in algebra one classes which thus comes at the expense of learning algebra. My daughter's class spent more than two weeks on this activity."
MD Algebra sample test Item #48is listed as a Data Analysis item on the state website. But, it is really a reading item, as noted in the Appendix
7.Bad Habits are Hard to Break.
Supposed a high school graduate was well trained in algebra and data analysis according to the problems on this test. Supposed that this graduate noticed that the price of some stock was $40 in January, $50 in February, $60 in March, $70 in April and $80 in May. Based on this type of training in recognizing, and extending patterns or fitting a "best line" to this data, the graduate will run out and use his/her savings to buy stock in this company, sure that its price will be $150 in December.
Math Educationer, Guershon Harel, has noted: "During the last two years, I have worked intensively with junior-high and high-school inservice teachers. One of the most robust behaviors I see with these teachers is the absolute reliance on pictures and patterns. Almost in all problems I give to the teachers, their first approach is to attempt to solve them by numeric patterns or by drawing pictures. Once the teachers observe a "pattern" (from two or three cases) or draw a picture, they derive the final conclusion empirically or perceptually. There are almost no attempts to support and supplement their empirical and perceptual reasoning with algebra".
Training students on such algebra and data analysis problems is training them to make rash, often incorrect projections about the future or invent incorrect formulas/shortcuts in Calculus. What is really important is the exposure/training in non-rigorous arguments. They can less readily see through the faulty reasoning so often presented in the media and by politicians. Also, they will have more difficulty adjusting to and understanding college courses.
(Emphasis in bold added.)
The MSDE web site states: "In all mathematics content standards, the emphasis is on achieving a balance among memorization of facts, proficiency with paper and pencil skills, appropriate use of technology, conceptual understandings, and problem solving".
"The Maryland Learning Outcomes -- Introduction/Rationale" states that "the core learning goals reflect a shift in emphasis from memorization of isolated facts and procedures and proficiency with paper and pencil skills to emphasis on conceptual understanding " (Granted that the traditional instruction of the 1980's sabotaged learning math by its over emphasis on memorization of isolated facts and procedures.)
This emphasis on conceptual understanding is purportedly exemplified by the "constructed response" items on the sample test, like Items #32 and 17 above.
The level of conceptual understandings and problem solving is very low,
often trivial on the sample MD Algebra test.
9.Baltimore Sun October 8, 2002
Set the bar high in math, ensure students clear it
By Donald N. Langenberg, Chancellor emeritus of the University System of Maryland, regents' professor of education K-12 and professor of physics and electrical engineering at the University of Maryland, College Park.
MARYLAND'S EDUCATION Department is fielding a new high school assessment (HSA) system, a set of end-of-course examinations intended to measure the performance of students in core subjects. The results are expected to be part of the high school graduation requirements for every student.
The presidents of all Maryland public universities and I signed a statement in 1997 supporting development of the HSA. It stated that "we have committed ourselves to align higher education admission requirements with the assessment standards and to incorporate the program goals into our preparation of the next generation of elementary and secondary teachers."
That's a worthy goal, but it is achievable only if the HSA standards are sufficiently high to ensure that any student who meets them is really prepared for success in college. That stark reality was illustrated by the recent Sun article about the dismal math literacy levels of many entering freshmen in our universities and the need for expensive and time-consuming remedial programs.
I recently worked through all 49 sample items posted on the Maryland State Department of Education's Web site for the forthcoming Maryland High School Assessment on Functions, Algebra, Data Analysis and Probability.
It's a pretty good test, but for two things: There is precious little algebra in it, and the level is too low. (A mathematician colleague has characterized the level as about sixth-grade. The course related to this test is taken by most students in the ninth grade.)
If the sample test accurately represents what is to come in the HSA, I fear that many high school graduates will be led to believe that they are good at math, but will require serious remediation in college. That is especially likely if they take similarly unchallenging courses throughout high school or are allowed to evade further exposure to any substantial mathematics.
We have heard the arguments for setting academic standards that most students can meet without much effort. They include the popular myth that only a few students have the special talent required for mathematics, so we can't realistically expect most students to learn it.
Recent international comparisons show that American students do about as well in math in elementary school as their foreign counterparts but steadily lose ground through middle and high school, finishing at the back of the pack. I doubt that adolescent hormones kill math ability. Rather, I suspect our schools fail to give math sufficient continuing emphasis, thereby making poor performance inevitable. Our students can't be expected to learn what they're not taught or what they are taught poorly.
Other foolish -- indeed, irresponsible -- arguments include the assertion that math literacy is unnecessary in most careers and that we might be embarrassed if our students perform poorly in math. Too many do, and we should be embarrassed. The first step in dealing with a serious problem is to acknowledge that it exists.
Then we must provide our students, their teachers and parents with an honest and accurate understanding of what it will take to succeed in tomorrow's world, and we must set the bar high and do whatever is necessary to help students surmount it.
That includes establishing school environments of consistently high expectations maintained by teachers fully capable of bringing all of their students up to internationally competitive performance standards. To do less is to deny our children access to important career paths and to resign our nation to also-ran status in today's global knowledge-based economy.
(MD Algebra test).
It is slated to be both
#1. a new high school graduation requirement and
#2. an end-of-year test Algebra test. It need not be both; originally it wasn't.
#3. "Each local system should use the [state test] to guide the development of their local mathematics curricula" (According to the state's website:). The Montgomery County math curriculum is currently being dumbed down in order to be aligned with the low state standards.
#4. The MD Algebra test will replace the MD Functional Math Test, which tests basic arithmetic calculations. Can the student multiply 23x37 by hand? The suggested replacement, the MD Algebra test tests whether the student can correctly punch "23x37" into a hand calculator. Being able to multiply 23x37 by hand will no longer be a HS graduation requirement
#5. "The presidents of all Maryland public universities and I signed a statement in 1997 supporting development of the HSA. It stated that 'we have committed ourselves to align higher education admission requirements with the assessment standards and to incorporate the program goals into our preparation of the next generation of elementary and secondary teachers.' There is precious little algebra in it, and the level is too low." Donald N. Langenberg, chancellor emeritus of the University System of Maryland, regents' professor of education K-12 and professor of physics and electrical engineering at the University of Maryland, College Park,. In Balt. Sun October 8, 2002
requirement from the end-of-year Algebra test
Far betterthat the MD Algebra test be only a graduation requirement entitled "The Maryland Assessment of Simple Math Problems".
Then most students could pass it in middle school and start Grade 9 with a real Algebra course as they do with the MD Functional Math Test.
The political reality is that graduation requirements must be low enough so that they do not prevent sizable numbers of students from graduating.
A course in Algebra is NOT a H.S. graduation requirement in MD. Only a course which includes some Algebraic concepts.
A separate end-of-year Algebra test would not have this restriction.
The current title is misleading -- a cover-up for the Algebra the students are not learning. We need a "Truth-in-Labeling Law" for state tests. Otherwise, parents will be misled into believing that their children have learned Algebra.
Appendix.Washington Post. Outlook,
the Sunday's Post's opinion and commentary section.
When It Comes To Math, Words Count
By Jerome Dancis
Sunday, September 8, 2002; Page B04
Last week, a report released by the Brookings Institution revealed that American students are falling behind in their arithmetic skills.
Given the demonizing of computational skills -- such as how to multiply 23 by 37 -- by the nationwide math reform movement (part of the educational reform movement, which demonizes memorizing anything), this was predictable. As an associate professor of mathematics at the University of Maryland, I've seen students arrive with increasingly poor training, and ever weaker math skills and knowledge. But computational competence is only a fraction of the problem. Another, largely unrecognized, factor is students' poor reading skills.
If Johnny can't read, then he can't do math. It is not enough to know how to add, subtract, multiply or divide numbers; one must also know which numbers to add, subtract, multiply or divide, and when to do which calculation. Students must be able to understand the wording of problems well enough to translate them into mathematical expressions and equations.
Having reviewed both the Maryland Functional Mathematics Test, which requires students to calculate, and its likely replacement, the Maryland High School Assessment on Functions, Algebra, Data Analysis and Probability (Maryland algebra test, for short), I know much of the math is absurdly easy. Yet students reportedly are doing badly on the field tests of the new algebra exam, even though they are provided with fancy graphing calculators when they take it.
Since students are given calculators, their poor performance cannot be blamed on trouble doing basic computation. So they must be having difficulty understanding what they are reading.
Two of the easier questions from the sample Maryland algebra test demonstrate this deficiency in reading comprehension. (You can view the sample test at http://www.mdk12.org/mspp/high_school/look_like/algebra/intro.html.)
QUESTION 1 (Item 2 from the sample test)
The United States Congress is composed of the Senate and the House of Representatives. The matrices [tables] below show the number of members in Congress from 1983 through 1989.
Senate 1983 198519871989
Dem.s. 54 53 55 55
Repub. 46 47 45 45
Indep. 0 0 0 0
House 1983 1985 1987 1989
Dem.s. 269 252 258 259
Repub 165 182 177 174
Indep 0 0 0 0
What was the total number of Democrats in Congress in 1985?
F 229, G 235, H 305, J 534
When Question 1 was field tested in 2000, one out of five students missed it. Most likely reading comprehension was the major reason for missing this question, since students can surely use a calculator to add 252 + 53.
QUESTION 2 (Item 48 from the sample test)
The table below shows how a typical household spends money on utilities.
Utility Pct. of Total Utility Costs
Water heating 14
Heating and cooling 44
A typical household spent $1,400 on utilities last year. If there are no significant changes in their (sic) utility usage for this year, how much should they (sic) budget for heating and cooling their (sic) home this year?
F $196, G $378, H $616, J $784
To answer Question 2, one would simply use a calculator to calculate 44 percent of $1,400 ($616). Yet, when Question 2 was field tested in 2000, 22 percent of students skipped it, and less than half -- 44 percent -- of students who tried it got it right.
This cries out for serious improvement in reading instruction. The arithmetic level of these problems is much lower than the reading comprehension level. And, overall, the math level on the Maryland algebra sample test is lower than the reading comprehension level.
Luckily, the Maryland legislature just passed a new educational funding law. Hopefully, some of the funds will be allocated to the training of our children in the very precise reading comprehension needed to do the math problems that arise in school and in the real world, as well as on tests.
Jerome Dancis, who has taught math at the University of Maryland for three decades, spent the past year advocating that Maryland put some real algebra in its state algebra test.