by Dr. Jerome Dancis (associate professor of mathematics, Univ. of MD, College Park
Comments presented to the MD State Board of Education, Oct. 28, 2003.
A child
of mine brought home test papers in Algebra I, Geometry and Algebra II with
mathematically correct answers marked wrong. (I was told that her teachers were MD certified math
teachers.) My impression is that
mathematically correct answers being marked wrong, is not uncommon in the state
of MD. One child of mine
asked her teacher how to do an SAT Math problem. The teacher obliged; but her calculations were completely
wrong. The teacher’s answer
was not even one of the four multiple choices. These incidents occurred at schools, which received national
Blue ribbons as schools of excellence.
The lack of knowledge of math
extends to those who wrote Montgomery County Public
Schools current Algebra I test; one, of its writers, had not a clue about
variables in Algebra (See Appendix 3, below)
That
many a Math teacher is not fluent in the math they are teaching is a natural
consequence of the low requirements for certification in MD. Of course there are many teachers, who
are knowledgeable.
In
fact, MD sets the bar for middle school math teachers at a lower level than
the bar of MD’s pretend[1]
Algebra test, a test aimed at weak Grade 9 students. This is because The Praxis II
Middle School Math Content Exam is being used for certification. (See Appendix 2, below) I suspect that the
largest concentrations, of weak middle school math teachers, are in schools
with large numbers of at-risk students.
(FYI,
my math department has two math courses (Math 210 and 211)[2]
designed for prospective K-5 teachers.
These four-credit courses are restricted to students, majoring in
elementary school education. We
have no course designed for prospective middle school math teachers; but
they can and do take the course designed for K-5 teachers.)
I have heard you board members
express concerns that far too many minority students scored below the passing
cut score on the MD’s pretend Algebra test and that appropriate
interventions be put in place in middle school to reduce the failure rate. In my opinion, four crucial
interventions would be:
I would
happily offer my services to assist in such training – except for the
parts of the test that are mathematically wrong and misleading.
In
addition to The Maryland Mathematics Commission slogan: “All students can learn
math”, we need my slogan “All math
teachers as well as all elementary and middle school principals and
vice-principals can learn math”.
Middle school teachers, who are fluent in advanced ratio problems, like Problem 1 in Appendix 2 below, will be able to provide useful instruction to their students, who will then be able to do Item 15 on the sample MD Algebra test. This will not be true for teachers, who are limited to simple ratio problems, like Questions 7 and 10 from the sample Praxis II test. (See Appendix 2 below.)
Appendix 1. Council of College Faculty Needed.
I was happy to read that Superintendent Nancy S. Grasmick wants to "listen to voices that haven't been heard." Hopefully there will also be a council of college faculty from departments of English, math, science and social studies – not just education. This is crucial for organizing high school education so that it prepares students for college. This is in contrast to The Maryland Visionary Panel for Better Schools, which contained no faculty member from my campus (the officially designated flagship campus of Univ. of MD), and no math professor from any college. The Maryland Mathematics Commission did have one math professor, who reported regularly being outvoted as a minority of one.
Two of the sample multiple choice questions, on the Praxis II Middle School Math content exam are "ratio" questions (# 7 and 10), both are simple ratio questions. None of the sample questions is an "advanced" ratio question like Problem 1 below:
Praxis II Sample Question 7. (An
simple ratio problem) (Abbreviated)
Suppose that 40% (by weight) of a county's trash is paper and 8% is
plastic. "If approximately 60
tons of the trash consists of paper, approximately how many tons of the trash
consists of plastics?"
Solution. The ratio of
plastic to paper is 8% / 40% =
1/5 or one to five.
So the
tonnage of plastics is one fifth that of paper or 60/5 = 12 tons.
Problem
1. (An advanced ratio problem) . ) Suppose that 40% (by weight)
of a county's trash is paper and 8% is plastic. If approximately 72 tons of the trash consists of paper and
plastic, approximately how many tons of the trash consists of plastics?
Note
that one has to stop a minute to think and analyze Problem 1, in contrast to
the straight-forward ratio calculation needed for the Praxis II Sample
Question 7.
Solution. Together,
paper and plastic compose 40% + 8%
= 48% of a county's trash.
The
ratio of plastic to {paper and plastic} is 8% / 48% = 1/6
or one to six.
So the tonnage of plastics is one sixth that of {paper and plastic} or 72/6 = 12 tons.
Remark. Item 15 on the sample Maryland High School Assessment on Functions, Algebra, Data Analysis and Probability (MD Algebra) test, is an advanced ratio problem. When Item 15 was field-tested in 2000, less than one third of the students had the correct answer – not much better than random guessing. Item 15 is on the MD Dept. of Education website at http://www.mdk12.org/mspp/high_school/look_like/algebra/v15.html.
Middle school teachers, who are fluent in advanced ratio problems, like Problem 1, will be able to provide useful instruction to their students, who will then be able to do Item 15 on the sample MD Algebra test. This will not be true for teachers, who are merely fluent in simple ratio problems, like Questions 7 and 10 from the sample Praxis II test.
Appendix
3: The lack of knowledge
of math extends to those who wrote
Montgomery County Public Schools
current Algebra I test
As Ken Sebens, professor of zoology, UMCP correctly noted: "but both problems [below] are mathematically incorrect. In the first, for example, if the number of band members changes from 64 to 60, it is not correct (in C.) to subtract 4 from b. b is a variable defined as the number of band members. b just changes from 64 to 60. The second problem has exactly the same error; n is the number of cheerleaders. It is not correct to add 2 to n in (C.)."
That
is, the writer had not a clue about variables in Algebra.
MCPS 1B EXAM REVIEW (Wash.Post, July 10, 2001):
The band is planning to raise $12,000. In order to reach their goal, the average amount that each band member must raise is a function of the number of band members, b, with the rule
f(b) = 12000/b
C. If four students drop out of the band then the function for the amount of money each student would raise changes to the function
g(b) = 12000/(b-4)
If instead five new students join the band, what new function rule would describe the relationship between the number of band members and the amount to be raised?
h(b) =
MCPS 1B EXAM:
The cheerleaders have been invited to march in the Cherry Blossom parade in Washington, DC. The total cost of the trip is $420. The average amount each cheerleader must pay is a function of the number of cheerleaders, n, with the rule
f(n) = 420/n
C) The original function f(n) = 420/n becomes g(n) = 420/(n+2) if two extra cheerleaders go on the trip. If instead, four cheerleaders DO NOT go on the trip, how will f(n) change?
Jerome Dancis’s e-mail address is jdancis@math.umd.edu. His website is www.math.umd.edu/~jnd. Read “Beware the MD Algebra Test” and “If Johnny Can’t Read, He Can’t do Math” therein.