Mathematics instruction in PGCPS
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.
A parent in
Montgomery County just emailed me this:
"Montgomery
County does not teach multiplication tables. That is something that parents are to teach at home. Unless things have changed,
multiplication tables were never part of the curriculum for the years 1996-2003
when our kids were in MCPS elementary schools."
Does your school system expect students to know what 9x8 is, or merely to have a
"sense" of what 9x8 is? Does your school system expect students to know the
standard method of calculating
23 x 37, or merely to be able to calculate
23 x 37 on a calculator?
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 www.gazette.net/stories/070407/prinsch174932_32360.shtml) 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
concernedctparent.blogspot.com/2007/07/when-reform-math-rubber-hits-road.html
(No www.) I recommend this.
Mathematics instruction in PGCPS
By Jerome Dancis
Topics
Notes from PG BOE's
curriculum committee's April meeting
PGCPS's
"LOOK-FORS" for Mathematics instruction
The HSA on Math
The "Math
Reform" movement and NCTM
SAT-PSAT Math
Notes from PG BOE's
curriculum committee's April 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.
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 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 http://www.all4ed.org/publications/ReadingNext/ExecutiveSummary.html,
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) (http://www.principals.org/s_nassp/sec.asp?CID=858&DID=52759)
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.
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.
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.
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, http://www.edweek.org/ew/1994/20lein.h13)
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
[http://www.mdk12.org/mspp/standards/math/introduction.html]: "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" [www.nctm.org/focalpoints/bygrade.asp].
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.
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.)
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.
Jerome Dancis, Ph.D. (math)
Associate Professor Emeritus
Math Dept, Univ. of Maryland
College Park, MD 20742-4015
Math Education Website: www.math.umd.edu\~jnd
301 345 2973 (c) 301 448 8132
email jnd@math.umd.edu
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
Email zandrab7@yahoo.com
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.
Sincerely,
Zandra R. Brown