Mathematics
instruction in PGCPS

**By ****Jerome
Dancis**

__Topics__

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

5. The "Math Reform" movement and NCTM

**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 CommissionÕs (MHEC) Student Outcome and Achievement Report
(SOAR).**

**Caveat**. *This particular
data *describes *only
those graduates of Prince GeorgesÕ 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

** **

**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
Ð 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: Òa [PGCPS] math coordinator [said] that county students
should
have a ÔsenseÕ of what 9x8 is.Ó 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 ProfessorsÕ 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**:
Ò**College ProfessorsÕ Concerns on
Mathematical Preparedness of Incoming College Freshmen**Ó is at http://sections.maa.org/mddcva/HS_students.php.
MAA is the professional association,
for
college math instruction, of college and community college professors
of
mathematics.

Quote
from Gazette at www.gazette.net/stories/070407/prinsch174932_32360.shtml.

**Jerome Dancis** is an Associate
Professor Emeritus, Math Dept., Univ. of MD. His related
articles,
Ò**Mathematics instruction in PGCPS**Ó
and Ò**Notes on Remedial Math Problem**Ó
and Ò**Comments on Statement on
Mathematical Preparedness**Ó are on his Math Education Website: **www.math.umd.edu\~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 MDÕs 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 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.

** **

**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 Ð 37, it is very useful for them to use dimes and pennies
(manipulatives). But, at some point students should progress to
subtracting 53 Ð 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 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 É *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 É * 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.

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! É "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) É 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.*

** **

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 MarylandÕs
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.**

É Encouraging
African-American students to take the
test without giving them the academic support to do well on it sets
them up for
failure É We cannot continue to encourage PSAT participation if
weÕre
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

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. EveryoneÕs 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
Òall the
timeÓ 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 ÒholesÓ in the studentsÕ 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