Mathematics instruction in PGCPS


By Jerome  Dancis



1.  Fewer PG Students Learning Arithmetic and Algebra

More Students Going to College, But Fewer Prepared to go

2.  PGCPS does NOT expect students to know how to calculate   23 x 37.

3.  PGCPS's "LOOK-FORS" for Mathematics instruction

4.  The HSA on Math

5.  The "Math Reform" movement and NCTM

6.  SAT-PSAT Math

7.  An elementary school mathematics specialist explains why the Math curriculum is unnecessarily hard for teachers to teach and unnecessarily hard for students to learn.


1.  Fewer PG Students Learning Arithmetic and Algebra

More Students Going to College, But Fewer Prepared to go


Analysis based on data by Maryland Higher Education Commission’s (MHEC) Student Outcome and Achievement Report (SOAR).


CaveatThis particular data describes only those graduates of Prince Georges’ County high schools in 1998, 2005 and 2006 who entered a college in Maryland the same year.  It compares those who were at least minimally prepared in Math -- not requiring remedial Algebra or Arithmetic -- vs. those who needed remedial Arithmetic or Algebra before being allowed to take college level math courses.  In reviewing the numbers, they reveal that the situation went from bad in 1998 to worse in 2005 and 2006 for all ethnic groups, but there were more dramatic downturns for African-American and Hispanic students.



Decline in Percent of PG HS Graduates Minimally Ready for College Math when they entered a college in MD. 


                                 1998         2005      2006

African-Americans          55%             37%       41%

Hispanics                          58%             39%       43%

Whites                               75%             66%       65%

Asian-Americans             81%             71%      72%


While downturns occurred in every ethnic group in the entire state of Maryland, the downturns for PG County were larger than for the state as a whole.  

Warning: The comparisons for PG white and Hispanic graduates going to MD colleges is problematic since their numbers changed considerably from 1998 to 2005 and 2006; the number of white graduates dropped from 669 to 476 and 502 and the number of  Hispanic graduates jumped from 140 to 209 and 195.


PG schools are becoming more successful at getting African-Americans into college. For females the number increased by 12%,  from  1257   to 1405 during 1998 – 2005, while for males the number increased by 23%,  from  874   to 1076. 


But at the same time, the number of African-American graduates who were minimally ready for college Math declined drastically.  For females, it dropped by 27%,  from  682   to 498.  For males, it dropped by 16%,  from  500 to  418.


For 1998 to 2005 and 2006, the total number of African-Americans going to college (in MD) rose from 2131 to 2481 and 2439; the number minimally ready for college Math declined from  1182  to  916  and  1010.  From 1998 to 2005, the number of African-American graduates who were [at least] minimally ready for college Reading (not needing remedial reading in college) dropped 2% (from 1453 to 1431).

Similarly, the number of Hispanic males going to college went up from  68 to 82 during 1998 - 2005, while the number of Hispanic males who were minimally ready for college Math, dropped, from  43 to 35.


One of the likely causes for the downturn:  High school Algebra I used to be the Algebra course colleges expected. Under the specter of the MD School Assessments (MSAs) and High School Assessments (HSAs), school administrators have been bending the instructional programs out of shape in order to teach to the state tests.  The MSAs on math and the MD Voluntary Math Curriculum marginalizes Arithmetic, thereby not allocating sufficient time for too many students to learn Arithmetic.  The MD HSA on Algebra avoids the arithmetic and arithmetic-based Algebra students will need in college, such as knowing that  3x + 2x = 5x  and  knowing  9x8 = 72.  


As Dr. Ronald Williams, (a vice president of the College Board and then president of Prince George's Community College [PGCC]) noted during his presentation, at the September 6, 2006 meeting of the Prince George's County Council's Blue Ribbon Committee on High Stakes Testing [State of Maryland HSAs]:

One eighth of the PGCC budget is allocated to remediation.  Specifically, there is a remedial math problem. Many students collect 40 credits at PGCC, but avoid the remedial math courses and then drop out.  Many other students just take remedial courses and then drop out.  There is a chasm between what students are learning in high school math and what PGCC demands (arithmetic and arithmetic-based Algebra).


As the PG Gazette noted:  “a [PGCPS] math coordinator [said] that county students should have a ‘sense’ of what 9x8 is.”  The implication being that students can use calculators to find that  9x8 = 72.  Having students use calculators is a good tactic if the only goal is students passing the MD HSA on [calculator-based] Algebra.  But, this reliance on calculators sets students up to need remedial Arithmetic and Algebra when they enter college.


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) broke tradition by issuing its first statement ever on the College Professors’ Concerns on Mathematical Preparedness of Incoming College Freshmen.  I paraphrase its key recommendation as:  Students should be able to perform basic calculations in Arithmetic and in Algebra, without the assistance of calculators.  This is the antithesis of the MD HSA on [calculator-based] Algebra.


Our children deserve a better instructional program.


Notes:   “College Professors’ Concerns on Mathematical Preparedness of Incoming College Freshmen” is at  MAA is the professional association, for college math instruction, of college and community college professors of mathematics. 

Quote from Gazette  at

Jerome Dancis is an Associate Professor Emeritus, Math Dept., Univ. of MD.  His related articles,  “Mathematics instruction in PGCPS” and “Notes on Remedial Math Problem” and “Comments on Statement on Mathematical Preparedness” are on his Math Education Website:\~jnd




2.   PGCPS does NOT expect students to know how to calculate   23 x 37.


It is now out in the open:  PG County school system does NOT expect students to know what  9x8  is, but merely to have a "sense" of what  9x8  is.  So the PG County school system along with the state of MD’s High School Assessments (HSA) on Algebra do NOT expect students to know the standard method of calculating   23 x 37,  but merely to be able to calculate  23 x 37  on a calculator.


Notes from PG BOE's curriculum committee's April, 2007 meeting:

A math coordinator said that it is sufficient if students have a "sense" of what say  9x8  is, implying that students do not need to know the actual number.  Fortunately, two of the board members took strong exception.  A math coordinator said that not all students can memorize the multiplication tables, implying that since some cannot none should be required to do it.   


A short version of my report Mathematics instruction in PGCPS (below), was picked up a reporter.  He ran with it writing: "Math guru critical of math curriculum" in the July 4 PG Gazette on the web at  Now I am a Math guru.

A nice-to-read blog based on this PG Gazette article is


"When reform math rubber hits the road..."

on the web at

(No www.)  I recommend this.



3.  PGCPS's "LOOK-FORS" for Mathematics instruction


The PGCPS's "LOOK-FORS" are check-off lists of items for administrators to observe (and to look for) when they visit classrooms.  All three "LOOK-FORS" lists for Mathematics instruction include this requirement:  "Manipulatives, math tools and calculators are readily available and utilized".  This is a very good strategy if the goal is just to have students pass the MD [calculator-based] Algebra exam. This is a counterproductive strategy if two goals are (*) to have students remember the multiplication tables and  (*) to have students avoid remedial Arithmetic and Algebra when they enter college.


Manipulatives are like training wheels for bicycles; they are good for beginners, but children should progress to riding the bicycles without training wheels and students should progress to doing Arithmetic calculations without  the aid of manipulatives.  For example, when students are first learning to subtract say  53 – 37, it is very useful for them to use dimes and pennies (manipulatives).  But, at some point students should progress to subtracting  53 – 37  without  the aid of dimes and pennies.  But students progressing to doing Arithmetic calculations without  the aid of manipulatives or calculators is NOT on the "LOOK-FORS" lists for Mathematics instruction, even in high school


The "LOOK-FORS" lists for Mathematics instruction emphasize form and test prep NOT learning.


The overuse of calculators allows students' Arithmetic skills to get rusty; it also covers up students' lack of fluency with Arithmetic



Please consider adding to the "LOOK-FORS" lists for Mathematics instruction:


(1).  When simple Arithmetic and Algebra calculations arise in a lesson, students should do the calculations by hand without  the aid of manipulatives or calculators

(2)   Teachers presenting the math in a manner that emphasizes understanding the Mathematics

(3)   Math lessons place Stress On Non-trivial Analytical Reasoning

 (4) Math lessons include non-trivial, multi-step Math problems

(5)   Instruction in reading comprehension and following directions, especially instruction for Math word problems.

(6)  When pedagogically useful, HW will include making connections with previously learned Math, by including an exercise which requires knowledge from both the lesson of the day and from  previously learned Math.

(7)  When pedagogically possible, HW will include an exercise which foreshadows (or is at least relevant to) the next lesson.

(8)   Students appear to be understanding the lesson.

(9)  All Math vocabulary words on the Math Word are defined correctly


An example of a Non-trivial, multi-step Math problem, which places some Stress On Analytical Reasoning appropriate for First grade might be:


Problem 1.  The price of a loaf of bread is two dollars.  The price of a gallon jar of milk is two dollars.  Johnny buys one loaf of bread and one gallon jar of milk.  He gives the cashier a five-dollar bill.  What is the change?


The suggested "LOOK-FORS" additions (2), (3) and (4) would be consistent with [previous] PGCPS Superintendent John Deasy's assertion that his staff believes that all students can achieve on a high level.


Support for (9):  All three "LOOK-FORS" lists for Mathematics instruction require that "There is evidence of interactive Math Word Wall OR Word Wall that includes Math vocabulary".   It is not uncommon for Math textbooks to have Math vocabulary words with incorrect definitions.  Recently, I viewed a Math Word Wall in Montgomery County, which contained several incorrect definitions. 


Support for Instruction in reading comprehension and following directions in Math class:


The report, "Reading Next:  A Vision for Action and Research in Middle and High School Literacy"  at, notes that: "Some 70 percent of older readers [between fourth and twelfth grade] require some form of remediation. Very few of these older struggling readers need help to read the words on a page; their most common problem is that they are not able to comprehend what they read."  This report strongly recommends literacy (reading and writing) programs for the bulk of middle and high school students; a crucial element of such a program would be:  

"Effective [literacy] instructional principles embedded in content [for example math class], including … content-area teachers providing instruction and practice in reading and writing skills specific to their subject area".  (Emphasis added.)


Similarly, a report published by the National Association of Secondary School Principals (NASSP) ( states:

"Historically, direct literacy instruction has been supported up to the third grade. However, there is a glaring need for it to continue so students can not only read narrative text, but also learn specific strategies to derive meaning from expository and descriptive text. When literacy instruction stops early, how can middle and high school students learn the strategies to read increasingly difficult text and to comprehend more abstract ideas?  If a regular student continues to need direct instruction to read and comprehend the text found in secondary textbooks, consider the tremendous need for instruction and intervention that struggling readers must require. And sadly, if students two to three grade levels behind their peers do not receive intensive literacy instruction, the results can be devastating because the struggling reader will not experience success within the content areas. Therefore, it becomes even more critical that secondary content area teachers better understand and teach specific literacy strategies to help students read and extract meaning from the written material used to teach the course content. Conclusions from the RAND Reading Study Group [2002] clearly support the need for continued literacy instruction at the middle and high school levels   …  * Secondary students in the United States are scoring lower than students in other comparable nations. This is especially evident as secondary students deal with understanding discipline-specific content text."  (Emphasis added.)


This NASSP report quotes a 1999 position statement by the International Reading Association, which argued for  "  * Highly skilled teachers who model and explicitly teach reading comprehension and study strategies across the content areas".

I have allocated class time to reading instruction for the somewhat complicated sentences and paragraphs, which come up in my college math courses.



4.  The HSA on Math


Here is a sample HSA Math problem, one that stymied more than  5  of  8  (65%) Grade 9 students, when it was field tested in Maryland.   This suggests that students found this problem to be more difficult than the average HSA Math problem


Problem 2.   (2000 sample MD High School Assessment Algebra test, Item #48)

"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 "


 [Students] need to reformulate the problem [as] "Find   44%  of  $1400".


The arithmetic level of Problem 2, is much lower than the reading comprehension level, since students had calculators to calculate  44%  of  $1400.  So it is reasonable to suspect that understanding the problem was a major reason for  5  of  8  (65%) Grade 9 students not solving this problem correctly.


In 2000, I served as the mathematics advisor for the California edition of Harcourt's Grade 6 math book.  I read all the sample questions for the Maryland HSA on Functions, Algebra, Data Analysis and Probability.  With minor modifications, the California edition of Harcourt's Grade 6 math book would be a good fit for at least 42 of the 49 questions on this sample Maryland High School test.   [The modification needed is two weeks of additional instruction on jargon and how to read (numbers off) graphs.  Maybe  4  out of the 49 problems are too sophisticated for Grade 6.  Questions #41 and 42 are too difficult for high school.


MSDE says that the Maryland HSA on Math was written for "all" students, hence, it is aimed at weak students.  This is why a third of MD students are passing the Maryland HSA on Math in middle school.  My bet is that many PGCPS TAG students could pass the exam in Grade 6.



5.  The "Math Reform" movement and NCTM

(The National Council of Teachers of Mathematics)


The PGCPS's elementary school Mathematics curriculum is based on the Maryland Voluntary State Curriculum for Mathematics, which in turn is based on The "Math Reform" movement and NCTM standards.   The bad effects, of adhering to the Maryland Voluntary State Curriculum, on Math instruction in a PGCPS elementary school are described by the school Math Specialist, Zandra R. Brown in her presentation at the May, 2007 meeting of the Maryland State Board of Education.  Ms. Brown's presentation is at the end.  It describes the specific situation in PGCPS, as such it complements this wordy section about the general "Math Reform" movement.


There are many professional Math educators who have low levels of expectations for students learning Arithmetic.  For example, Steven Leinwand, who was the co-chairman of the U. S. Dept. of Education's Expert Panel (on Math textbooks) and Connecticut's Department of Education's State Coordinator of Math and was on the Board of Directors of The National Council of Teachers of Mathematics (NCTM).  He wrote:  "It's time to confront those nagging doubts about continuing to teach our students computational algorithms for addition, subtraction, multiplication, and division [like 23 x 37].  It's time to acknowledge that teaching these skills to our students is not only unnecessary, but counterproductive and downright dangerous! … "Today, real people in real situations regularly put finger to button [on calculator] and make critical decisions about which buttons to press, not where and how to carry threes into hundreds columns."   (Education Week on the Web, February 9, 1994,

A "Reform" movement of professors of mathematics education largely organized and wrote The National Council of Teachers of Mathematics (NCTM) Standards in 1989.  The NCTM is the professional society of school mathematics teachers.  This "Reform" movement demonized memorization of facts or proficiency with paper and pencil skills.  The 1989 NCTM Standards state: "This is not to suggest that valuable time should be devoted to exercises like  (17/24) + (5/18)".


This "Reform" movement stresses over-arching themes from K-12.  In Math, the over-arching themes are something like Arithmetic, Algebra, Geometry, Measurement, probability, Data analysis and problem solving. With so many topics to teach each year (in K-8), there is no way to have a coherent curriculum.  Also, soon after a topic is started, it is time to move on to the next topic; this occurs before the learning is moved into long-term memory.  Also far too little time is allocated to Arithmetic.


In 2000, NCTM issued its revised standards, 'Principles and Standards for School Mathematics' (PSSM).  Theses standards were an improvement, but still bad.  They did not demonize Arithmetic; but only marginalized Arithmetic.   The MD Math state Math curriculum has copied this marginalization of Arithmetic, the result is insufficient class time allocated to Arithmetic.


(From the Maryland State Dept. of Education web site  []: "The Maryland Mathematics Content Standards (Standards) …  are closely aligned with the National Council of Teachers of Mathematics (NCTM) 'Principles and Standards for School Mathematics' (PSSM).


In 2006, NCTM partially changed emphasis, when it issued its "Curriculum Focal Points for Mathematics in Prekindergarten" [].


School districts, Math textbooks, and state exams, which adopt these Focal Points, will greatly increase their emphasis on Arithmetic and greatly decrease their emphasis on superficial Data Analysis and Probability.  YEA!   The PGCPS would be wise to sign onto the NCTM Focal Points, the latest and best NCTM view of Math education.


Unfortunately, even these better NCTM Focal Points have quite low expectations when it comes to Arithmetic word problems.  Let's relook at:


Problem 1.  The price of a loaf of bread is two dollars.  The price of a large jar of milk is two dollars.  Johnny buys one loaf of bread and one large jar of milk.  He gives the cashier a five-dollar bill.  What is the change?


I would consider instruction for Problem 1, to be appropriate for Grade 1.  But, the new improved NCTM Focal Points considers instruction for Problem 1 to be appropriate for Grade 6.  This is typical of the low standards on analytical reasoning of the Math Reform movement, despite its claims to stress analytical reasoning.


The NCTM and the popular Math Reform curriculum emphasizes wordy "real world problems", usually with little math content, for example reread Problem 1, above. Maryland's Algebra exam exemplifies this, for example reread Problem 2, above.




6.  SAT-PSAT Math


The PGCPSS is advising grade nine students to take the PSATs.


Please note this warning from the report of the "Task Force on the Education of Maryland’s African-American Males":


"Increase the proportion of African-American males taking the PSAT in 10th grade and provide them the academic preparation and support they need to score well on it.

…  Encouraging African-American students to take the test without giving them the academic support to do well on it sets them up for failure …  We cannot continue to encourage PSAT participation if we’re unable to improve performance, for raising expectations only to dash them is a cruel compromise."


Let's look at a typical SAT Math problem, one that the SAT rated as a medium level problem.


An SAT medium level Problem. "How many minutes are required for a car to go  10  miles at a constant speed of  60  miles per hour?"  (Item#5 of Section 7 of the May 2000 SAT Math test.) 

 (Solution.  {60  miles per hour}  is  {a mile a minute}, so ten minutes needed to go 10 miles.)

Instruction for the many SAT-PSAT Arithmetic problems belongs in middle school.  It would be inappropriate to include instruction for such problems in an Algebra I or Algebra II course. SAT-PSAT Arithmetic problems are two-minute problems; each would be an ideal warm-up problem in middle school.



Our children deserve better.



 7.  An elementary school mathematics specialist explains why the Math curriculum is unnecessarily hard for teachers to teach and unnecessarily hard for students to learn.


Math Specialist, Zandra R. Brown's presentation follows.Zandra R. Brown

7 Maplewood Court

Greenbelt MD 20770-1907

Phone 301-513-5996

Home Phone 301-441-3138




May 30, 2007



Maryland State Board of Education

200 West Baltimore Street

Baltimore MD 21201



Dear Members of the Maryland State Board of Education,


I am writing to you on behalf of my students, my teachers, and myself regarding the current Maryland Voluntary State Curriculum for Mathematics.  I am a Math Specialist who works with teachers and students in an elementary school that houses Kindergarten through fifth grade.  We also have students who range in physical and academic abilities from multiple handicapped to talented and gifted status.  I have been blessed to be a teacher in such an environment with staff who constantly strive to meet the needs of their students and an administration that tries to balance the dictates of the school system with the input of the staff.


I began my teaching career during the time of MSPAP.  I was ecstatic when it was no longer a viable test to be used for measuring student progress according to the No Child Left Behind requirements.  The days of trying to have the correct groupings so that every student might be successful on the MSPAP were gone, but we all dreaded what would take its place.  Everyone’s notion of accountability standards and measureable objectives could be different from state to state.  We knew that Maryland would set the bar relatively high because the state always wants to be one of the leaders in education.  When the Voluntary State Curriculum was released, many teachers in my school were taken aback by not only the number of indicators to be covered in Mathematics but also the number of different areas.  What looks like a wonderful, comprehensive document for assessing the Mathematical knowledge of our students is actually a nightmare when it comes to the logical building of a solid Mathematics foundation.


The Voluntary State Curriculum calls for students to be proficient in seven conceptual areas of Mathematics and each area then has multiple indicators.  This is something you are familiar with but you may not be familiar with the instructional impact and its implications for our students.  There are competing philosophies regarding the teaching of Mathematics to young students.  Some philosophies believe children should discover answers to Math situations by being given the basic tools but never direct instruction for how to obtain the answer.  Other philosophies believe that rout memorization of all basic facts will solve all the computational problems being experienced by the intermediate grades because if students just knew their facts great things could happen in Math.  All philosophies seek to make our children better in Mathematics but how often do you listen to the teachers of the children?  The answer may be “all the time” but do you really hear them as I do?


I love Mathematics but I hate how we are forced to teach it because of the Voluntary State Curriculum.  I thought for many years that I was one of the only teachers who felt that teaching Math in this manner did not make sense.  Why are we hopping from one skill or concept to another like a rabbit looking for greener pasture?  Are we hoping to find something our students are successful at doing within Math so we keep changing topics?  These are questions that I have heard from other teachers and they echo my own feelings.  In order to meet all the requirements of the VSC, our students never fully have an opportunity to understand any one concept or skill.  One moment they are having to understand what a fraction is and the next they are adding decimals.  August may be solving algebraic equations and January is multiplying decimals.  There are natural progressions that can occur amongst these concepts but those progressions have to be sacrificed in order to teach all of the required indicators.


While the schedule is left to the individual school systems, the VSC drives the instructional objectives for the year.  The VSC needs to be changed.  It needs to be re-evaluated and some dawning realizations need to occur.  1)  Everything does not need to be covered every year.  If you thoroughly cover addition and subtraction in first grade, then students will have that skill whether they are adding or subtracting decimals or figuring out perimeter.  The skill will get used time and again but a firm foundation needs to be established so students have a better sense of number relations.  2)  Fewer but more thorough teaching of concepts is better.  This harkens to the recommendations set forth by the National Council of Teachers of Mathematics in the Focal Points.  The third graders really understood fractions.  They could compare them.  They could put them in order.  They could add them and subtract them.  Guess what?  They probably could have learned how to multiply them also since they learned multiplication this year too and the teachers would have loved to have used the opportunity to teach the students about using fractions to measure things but they had to move on to comparing decimals.  You would have to understand that they had already covered Probability, reading and interpreting graphs, adding and subtracting across zeros and 2- and 3- digit numbers, geometric shapes both 2-dimensional and 3-dimensional, place value to the thousands, and the list goes on but next on the list is decimals so no more fractions.  And lastly, 3) we have had 5 years of the VSC and things are getting worse not better.  This is a change that needs to happen at the state level and it needs to start now.  While test scores on the MSA may rise, it is not an indicator of more knowledge but an indicator of better abilities to teach to the test.  You would be amazed at the “holes” in the students’ understanding of Mathematics, but the teachers in the classroom would not because there is only so much time to teach 66 indicators in 7 concept areas to a group of third graders.


Thank you for your time.



Zandra R. Brown