College Readiness -- A Simple Description
By Jerome Dancis, Associate Professor Emeritus, Math Dept., Univ. of MD
Math Education Website: www.math.umd.edu/~jnd
Ready for college. To survive academically the first year of college, students basically need the three Rs, Reading, wRiting and aRithmetic, albeit all on high school levels. No Statistics needed.
Reading means reading with understanding the expository and descriptive text in science and social studies textbooks, not literature. This includes following written directions. Writing means writing a coherent summary of each chapter in the science and social studies textbooks, and relating the chapter to material previously studied. Arithmetic means Arithmetic, including fractions, decimals, percents, measurement and multi-step Arithmetic word problems, along with "generalized" Arithmetic, better known as Algebra.
Low level College-ready Math Standard (Ready to enroll in a credit-bearing Math course; this is supposedly Common Core's goal). Graduates should be fluent in Arithmetic and real (1980's) high school Algebra I, without calculators.
College freshmen, not knowledgeable in Arithmetic or real high school Algebra I, are relegated to remedial math courses; colleges are not real successful at teaching these courses. Some might considered it unfair to require Grade 8 students to be able to add fractions, when MSDE does not require this for its endorsement for "highly qualified" middle school Math teachers.
As our 40+ Mathematicians' public letter, "RACE TO THE TOP AND K-12 MATHEMATICS EDUCATION" says:
For the United States to remain competitive, every part of K-12 mathematics education in this country must be strengthened: curriculum, textbooks, instruction, assessments, and, above all, the preparation and continuing professional development of those who teach mathematics and science, regardless of grade level and the kind of school in which they teach.
All prospective K-8 mathematics and science teachers, coaches, and supervisors should be required to pass a solid test on the core mathematical material (especially arithmetic) for licensing. Mathematics supervisors and coaches should be required to have at least the mathematics qualifications of those they supervise.
We should follow Massachusetts's example of requiring aspiring elementary school [Grades 1-5] teachers to pass a math-specific test to earn their teaching license.
We need content-rich professional development programs for current K-8 mathematics and science teachers, coaches, and supervisors, and for elementary and middle school principals.
Fully college-ready Math Standard: To be ready for any Science, Technology, Engineering and Mathematics (STEM) major in college a graduate needs to be fluent in Pre-Calculus. This, in turn requires fluency in Arithmetic and Algebra II. A grade of C is not sufficient; depending on curriculum and teachers' standards, a grade of B (or even A) may not be sufficient. My guess is that a score of 600 on the Math SAT and on the SAT II advanced math exam are necessary, but not sufficient, for success in college calculus (for engineers).
Adding the Arithmetic and Pre-Algebra Math SAT and PSAT questions to the middle school Math curriculum would be a good step toward making all students more college ready. This would also make 600 a reasonable goal for the average score of MD graduates on the Math SAT. The SAT rated the following problem a 3 for difficulty on its scale of 1 to 5. Instruction for such problems usually is not included in the Math curriculum.
A mid-level SAT Problem. "How many minutes are required for a car to go 10 miles at a constant speed of 60 miles per hour?"
Also, competency with spread sheets and graphing calculators would be useful. (Not just type in a function and the hit graph.)
Probability and Statistics. There are snippets of data analysis, which would be useful for college freshmen. These include knowledge and understanding of means (averages), medians, deciles, box and whisker diagrams; also being able to read and draw a variety of graphs, charts and tables.
Warning. Unfortunately, Probability and Statistics is a major strand in the middle and high school part of the March draft of the Common Core Math Standards and in the Maryland (Voluntary) State Math Curriculum. The 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 knowledgeable in Arithmetic and Algebra.
For example, on my campus (UMCP), 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".
Thus, Probability and Statistics are NOT useful for college readiness. Freshmen, with a 5 on AP Statistics exam, and a C in high school Pre-Calculus, wishing to major in statistics (in college) will be at-risk in this major. Freshmen, with zero K-12 Statistics, but aced high school Pre-Calculus are fully ready to major in statistics. (The first college Statistics course, for students majoring in engineering or statistics, is based on calculus; it makes AP Statistics look like child's play.)
High school ready in Math for rigorous high school chemistry and physics classes requires fluency in Arithmetic including (*) measurement and (*) multistep word problems, as well as on (*) fractions, decimals and percents and on (*) units and proportions. Also required is automaticity on decimal equivalents of percents and fractions.
NOT in the March draft of the Common Core Math Standards for middle school.
High school ready in Math for high school personnel finance classes requires fluency in decimal equivalents of percents and fractions.
NOT in the March draft of the Common Core Math Standards. For example, memorizing that 50% equals a half, or even being able to figure out that 50% equals a half or .5 are NOT included.
In sharp contrast: "As his [Grade 5] students lined up after lunch outside his classroom, he popped questions before they could enter. 'Kayson, what is two-fifths as a percent?' he asked. The boy hesitated before correctly answering 40. 'Next time,' Mr. Skeeter said, 'quicker.'"
College math professors are distressed by the low level of understanding of arithmetic and arithmetic-based Algebra by masses of college students. This is why the MD/DC/VA SECTION of the Mathematical Association of America (MAA) has broken tradition by issuing its first statement ever on the College Professors' Concerns on Mathematical Preparedness of Incoming College Freshmen [www.salisbury.edu/mddcvamaa/HS_students.asp]. I paraphrase its key recommendation as: Students should be able to perform basic calculations in Arithmetic and in Algebra, without the assistance of calculators. Thus, the Algebra I, needed for college, is largely excluded from the MD HSA on Algebra.
Also see my reports "Comments on Statement on Mathematical Preparedness" as well as my "Notes on Remedial Math Problem" and "A Review of the Report of the Task Force on the Education of Maryland's African-American Males -- Useful Initiatives that should have been included" on my Math Education Website.
Goal for English classes Grades 4-12 should be that students can understand their science and social studies textbooks and be able to write a coherent summary of each chapter (one page or less); this includes relating the chapter to material previously studied. These are summaries, not outlines or reviews. Students need to be able to paraphrasing what a teacher has said. Reading includes paraphrasing a word problem (from science or math), accurately and precisely, into mathematical expressions, formulas or equations, as well as the reading of tables, charts and graphs.
This would require replacing perhaps half of the literature in these English courses with paragraphs from their science and social studies textbooks. Proficiency in literature is important, but it is not necessary for college readiness.
My impression is that high school teachers are required to pass a course on teaching reading in the content areas, but on my campus there is a single generic course, but reading history is very different from reading Mathematics. What is needed is that high school math teachers are required to pass a course on teaching reading specifically for Mathematics; similar for other subjects.
Writing and Speaking. Student need to be able to write and speak paragraphs coherently, clearly, concisely, comprehensively, logically, accurately and precisely without being cryptic, vague, ambiguous, obscure, redundant or repetitive. Unfortunately, for students, the writers of math textbooks do not model such writing, (perhaps true for other subjects). Such clear writing needs to be made a main focus of English and social studies classes in Grades 4-12. Also, a focus of science and mathematics classes.
Also see: "Literacy (writing and reading) is crucial" on my website at www.math.umd.edu/~jnd/Literacy.htm.
Our children deserve a better instructional program.
"Along a wall [of the Dundalk campus of the Community College of Baltimore County's Student Success Center] is a rack of handouts explaining points of grammar that might have last been explicitly taught in middle school, a measure of the immense ground to be made up. One covers comparative adjectives, explaining 'more' vs. 'most' or 'smarter vs. 'smartest.' Another discusses using pronouns and verb tenses." ["At 2-Year Colleges, Students Eager but Unready", N. Y. Times, Sept. 2, 2006 www.nytimes.com/2006/09/02/education/02college.html?_r=1&sq=McKusik&st=nyt&scp=1&pagewanted=print]
Ezra Shahn wrote ["On Science Literacy," in Educational Philosophy and Theory. Journal of the Philosophy of Education Society of Australia, (1988).]: "In descriptions of many biological phenomena " 'understanding' means mastery of a sequence such as A then B then C then D " . It was as though in reading or hearing 'then' the student was understanding 'and'. " [But] the sequential relationship is more restrictive, hence more precise and it is this distinction that many students apparently fail to grasp." Shahn also wrote: "it seems that students often misread conjunctions [including the implication words 'because' and 'then' (as in 'A then B')] so that they mean 'and'. "
Arnold Arons [A guide to introductory physics teaching, Chapter I Underpinnings, Section 1.16 Language] wrote: "... essentially the same problem frequently arises in connection with "if & then" statements of reasoning." Arons elaborated: "Crucial to understanding scientific reasoning and explanation [in beginning physics classes] as opposed to recall of isolated technical terms, resides in the use of [implication words] words such as "then" and "because".
From a Univ. of MD, College Park [UMCP] instructor:
I ... would like to comment in light of a recent quiz I gave in STAT 100 [Univ. of Maryland, College Park version of AP Stat]. Students were given a data set of five ordered pairs and asked to calculate the least squares regression line. Most all obtained the [correct] equation y = .52 + .7x, but many (over half of my class) missed points because they did not correctly graph the line. Interestingly, many students omitted the scale of the graph, and even though most all plotted the ordered pairs, the line they drew on the graph was not y = .52 + .7x, but some arbitrary line. When I returned the quiz and was talking to my classes about this, a comment I received was, "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 on the quiz paper for the student to receive credit, the reply was, ".... the calculator doesn't show the scale so I don't know how to do that." (By-the-way this was an assigned homework problem from the text.)
Another question on the quiz asked students to predict the volume of a tree with a diameter equal to 20 given that the equation of best fit by least squares for the diameter vs. volume was y = -36.9 + 5.07x. I had more than one student who forgot a 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--I reviewed how to graph a line in class (Algebra I) which the text assumes students know before taking STAT 100.
End email from UMCP instructor.
Relatedly from Homeroom Column of Montgomery Extra Section of Washington Post, December 25, 2003; Page GZ06: "With 'Pretend' Testing, a Poor Imitation of Preparing Students":
And I [Chenoweth] would like to know that kids can graph a simple line without a graphing calculator before they graduate from high school.
The head of math instruction for the state, Donna Watts, disagreed. "The technology is there. It's not going to go away," she said. "There is a limited population who can do math symbolically, the way mathematicians do. If this is an exam for all students, we want to make it comfortable for however students learn."
Problem. You borrow $100; you agree to pay ten percent interest. How much interest do you pay?
A Blair High School [Montgomery County] teacher of personal finance wrote me: "... I would not be at all surprised that it would stymie 65% of my kids doing the problem with simple numbers and no calculator."
Towson State University Professor of Biological Sciences, Virgina Anderson provides extensive training to her college biology students in the reading and drawing of tables, graphs and charts. The general need for such instruction in college is indicated by a federal study conducted by the National Center for Education Statistics. " ... 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 not designed to test your understanding of Proust, but to test your ability to read labels,' [Mark S. Schneider, commissioner of education statistics] added." ["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]
Just as fast as students can change 236 cents to $2.36, they should be able to change 236 centimeters to 2.36 meters and 236 percent to 2.36.
To achieve this, the state standards for certification need to be raised. Note:
"When Grant Scott, a biology teacher, had to teach (his chemistry students) at Howard High School, Howard County, MD, how to change centimeters to meters, he just told them to move the decimal two places -- rather than illustrating the concept. ... 'Forty-five minutes later, only three of them got it.' " [Washington Post, February 15, 1999]
The Arithmetic part of a two-line calculation in a Chemistry textbook [Zumdahl, 5th edition]: Simplifying the following messy fraction [where "^" denotes an exponent]:
(1.0 x 10^1) (1.0 x 10^2)^2
---------------------------------------- = 4 x 10^5.
Students should be able to simplifying this quickly, without a calculator, mostly doing mental math. This type of Arithmetic calculations is not part of the high school math curriculum; it belongs in middle school math, then it will be reviewed and practiced in high school science classes.
Data analysis is often tricky. Too tricky for high school and too tricky for the writers of the MD HSA on [Some concepts from] Functions, Algebra, Probability and Data Analysis. Let's look at 2007 Public Release Algebra/Data Analysis Item #38 at
This is also Item #37 at
"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?"
To obtain the correct answer, 2,112, students are expected to make the following unwarranted and usually incorrect assumptions:
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".
None of the people surveyed gave wrong answers, because it would be embarrassing to publicly say how they were really planned to vote. This is often not a true statement.
The fraction of people who will vote "No" in the next election is exactly (not just approximately) the same as the fraction of people who voted "No" in the survey.
The writers of the state math exams appear to have had little or no training in how to write Math precisely, without ambiguity and without loopholes. Writing Math correctly shares these attributes with writing law correctly.
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) doing her HSA "Algebra" homework on "best fit" lines:
"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."