Statistics, Percents, Rational
Numbers, Measurement, Geometry, Algebra, Probability: these are the units
that are taught in the Math B course in Montgomery County, Maryland. From
these topics the teachers devise the lessons in which the students will
accomplish the objectives put forth by the county. Yet, where does the
county come up with these unit topics? Most likely they are following the
Principles
and Standards document written by the National Council of Teachers
of Mathematics. Founded in 1920, NCTM is the world’s largest mathematics
education organization with more than 100,000 members in the United States
and Canada. NCTM is an organization whose mission is to provide vision
and leadership necessary to ensure a mathematics education of the highest
quality for all students. Principles and Standards, published in
2000, provides guidelines for excellence in mathematics education and a
call for all students from pre-kindergarten through grade twelve to engage
in more challenging mathematics. The document reflects educational research;
society’s needs to improve mathematical literacy; and observations by classroom
teachers, teacher educators, educational researchers, and mathematicians.
The document includes students work, classroom examples, and examples in
order to explain points made in the text. This organization has had substantial
influence as a guide to building instructional programs in our school systems.
As in most educational reforms, there is always another organization that believes their way is better and more beneficial for all. This particular organization is Achieve Incorporated. Achieve’s vision is for all students to be mathematically proficient in order to fully participate in the 21st century. Achieve’s Mathematics Advisory Panel (MAP) is made up of classroom teachers, mathematics educators, mathematicians, and curriculum developers. MAP recognizes the importance in improving student performance on many levels: supporting teachers, measuring student’s performance, and using assessment to improve classroom practice. The document Foundations for Success: Mathematics for the Middle Grades identifies the knowledge and skills that students need to develop to be successful, before entering high school. MAP’s work is based on the results from the Third International Mathematics and Science Study (TIMSS) and the follow-up study (TIMSS-R), which describes how U.S. students perform when compared with their peers from around the world. The data shows that in math classrooms in grades six through eighth simply repeat concepts that have already been taught and does not study any topic intensely. Also, the curriculum in the middle grades mostly focuses on arithmetic computation (Foundations for Success, 2001). Foundations for Success offers guidelines to provide school systems a mathematics education that is based on the countries that are rated the best in the world. Included in the document are sample problems and solutions that help illustrate the meaning of the expectations. These should be used in classroom discussion, not for assessment purposes. This document is a draft and has not been implemented yet.
Each document presents a series of
objectives in which, according to them, all students should be able to
know and understand upon the completion of each unit or strand. In both
documents some of the objectives set forth are reasonable for the students
at the age they are recommending and some are not going to be easily attained
by all students. On reason is the level of development a student is at
each stage in their schooling will not be the same of that of their peers.
Also, in the Achieve document most of the sample problems will be too difficult
for a student in the middle grades to accomplish. More specifically, students
in the middle grades, 6-8, will have difficulty achieving the expectations
and standards set forth by NCTM and Achieve in the Data Analysis standard
and Data strand, respectively.
This is the first objective NCTM
expects all students in grades 6-8 to accomplish before entering high school.
This statement alone would probably be achievable among all students at
some point within grades 6-8. However, the council goes on to make more
specific goals under this heading of data collection which significantly
raises the bar in the expectation of the students accomplishing this objective.
The first is, "all students should formulate questions, design studies,
and collect data about a characteristic shared by two populations or different
characteristics within one population" (National Council
of Teachers of Mathematics, 2000, p. 248). To me, these are two different
objectives; one is analyzing two populations and another is describing
one population- two completely dissimilar objectives. Moreover, in the
Principles
and Standards document, the example corresponding to this objective
discusses a teacher asking her class to examine how different characteristics
in the design of a paper airplane can effect how far it travels. The different
characteristics are putting one paper clip on the nose of the airplane,
then using two paper clips. Then having her students’ design an experiment
in order to collect data to examine the effects the different designs have
on the distance the airplane travels. In my opinion, this is an adequate
example of the objective they want the students to be able to perform.
I also think that this would be a terrific lesson for a class of high achieving
students. In my teaching experience I have worked with students of all
levels, most recently, low achieving students and students with special
education services. These students would not be able to design the study
and then collect the data without me designing it for them and giving them
a lab sheet in which to record their results. It’s not that my expectations
are low or that I don’t think they are capable, I know what little amount
of independence they have had in their mathematics experience. Accomplishing
the "designing studies" aspect of the objective is asking a lot of these
students, who for most of their schooling has been pushed on in the system
and hand-held by their teachers. After much guidance, eventually the students
would be able to accomplish the goal; as a result they would forgo other
objectives because of the time factor to complete this particular one.
The other aspect of this first objective
mentioned by the council is, "all students should select, create, and use
appropriate graphical representations of data, including histograms, box
plots, and scatterplots" (NCTM, 2000, pg. 248). This
objective is absolutely appropriate and the students at this level of schooling
will be able to accomplish this goal. My teaching experience has been in
a seventh grade classroom for four years in two different schools, I have
taught each of these graphical representations each year and all of my
students succeed. They realize the importance and the purpose of each of
the different graphs and when it is appropriate to use each graph. They
also learn to analyze the graphs by examining measures of central tendency,
making predictions, and finding misleading statistics.
The second objective in the Data
Analysis and Probability standard for grades 6-8 enables the students to
examine characteristics of the data that is represented in the graphs.
Under this heading, there are two goals that the council suggests that
all students should be able to accomplish. The first, "find, use, and interpret
measures of center and spread, including mean and interquartile range"
(NCTM, 2000, p. 248). This is another goal that is
attainable for students at this age. In the Principles and Standards
document, the council focuses on students differentiating between the mean
and median. A misconception among many students is that the best way to
describe a set of data is to use the average number, or the mean, without
realizing when it is appropriate to use the mean or median. The students
will learn when it is best to use a measure of central tendency based on
the distribution of the data in the set. They will also examine how these
measures change when pieces in data are added or deleted. My students have
a number of experiences finding the mean and median; changing the data;
and analyzing what happens to the mean and median as the data changes.
It is an important task for teachers to guide their students in effectively
demonstrating how changes in the data values affect the mean and the median.
The use of electronic software is also mentioned as a helpful tool for
the students to create a better understanding of the appropriate uses of
mean and median. This seems to be an instructional approach that would
be very interesting to our students, being that most of them have computers
at home. However, I implemented this approach in my classroom and did not
get the response from the students that I was expecting. The program presents
how the mean and median change when data is added or deleted in a context
that is engaging to the students. It allows the students to see that the
mean and median serve different purposes and that while the mean will always
change, the median may stay the same unless the data is moved over the
middle of the data. I thought this program would be an excellent way for
the students to interpret measures of center, unfortunately, some still
did not understand why the changing of the data points changed the mean
and median. I believe that most students in this grade band should understand
this concept, but for NCTM to maintain that all students should be able
to accomplish these goals is unreasonable.
The second component of this objective
is, "discuss and understand the correspondence between data sets and their
graphical representations, especially histograms, stem-and-leaf plots,
box plots, and scatterplots" (NCTM, 2000, p. 248).
Basically, this is asking the students to be able to make comparisons between
different graphs that are representing the same set of data. Previously,
I taught a class of eighth graders in which most of the students were low
achieving. These students, for example, were not able to notice clusters
in the data on one graph and find the median on another graph and make
correlations between the observations. These observations will be able
to be obtained by higher achieving students, but to believe that all students
can accomplish this goal is unrealistic.
This goal is broken up into three
parts. The first, "use observations about differences between two or more
samples to make conjectures about the populations from which the samples
were taken" (NCTM, 2000, p. 248). The students in grades
6-8 are capable of making observations from a graphical representation,
and comparing their findings with other data from the same graph. The students
should be able to understand how to compare and describe two or more data
sets within one graph using the appropriate vocabulary. I was able to do
this successfully with my students. Given data on three player’s free throws
in a season, displaying the three sets of data on a box plot. They were
able to distinguish which player had the largest range, interquarile range,
and median. As an extension, they wrote a journal entry describing if they
were a coach, who they would pick for their team and why. They knew not
to pick the player with the largest range because of his or her history
of shooting a low number of free throws. Most of the students chose the
player that had the highest median score. The example described in the
standards document compares the box plots from the paper airplane data
using one paper clip, then two paper clips. The students can make inferences
and pose questions about the data and what effects the paper clip had on
the paper airplane in terms of the distance traveled. This part of the
objective I have just mentioned is not only attainable for students in
grades 6-8, it is useful for them in their daily practices.
The second section, "make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit" (NCTM, 2000, p. 248). This another objective I agree with at this grade level that all students can accomplish. When constructing scatterplots, students must draw a trend line or a line of best fit. In the middle grades, the line should be approximate, just to show the trend. From this line the students can make conjectures about the relationship between two characteristics of a population. This also allows them approximate points on the line, and pose new questions. As an extension, the students can recognize this as a linear relationship, as well as positive, negative, or no relation. In my experience, I have noticed interpreting the scatterplot already graphed is not the challenge, but that the problem lies in deciding which is the independent variable, which is the dependent variable, and graphing it correctly. At this age, students have difficulty deciding which is the cause and which is the effect. However, I do agree that this goal is achievable.
The last goal of this objective,
"use conjectures to formulate new questions and plan new studies to answer
them" (NCTM, 2000, p. 248) is a challenging one. It
has been established that students can make conjectures or conclusions
about findings in the data and formulate questions based on the data. This
objective is asking the students to conduct their own study that would
prove that their conjecture holds true for a larger population. The example
used in the document suggests that if a sample of the population of the
sixth grade is tested, for instance, one sixth grade class, could the results
from this study hold true for all sixth graders in the school? Town? Country?
I think it would be a difficult task for the students to implement a fair
study that would elicit the results they are looking for in a population
much larger then they can even imagine.
The last objective in the Data
Analysis and Probability Standard for grades 6-8 is broken up into three
sections. The first expects all students to, "understand and use appropriate
terminology to describe complementary and mutually exclusive events" (NCTM,
2000, p. 248). It is reasonable to ask the students to use the appropriate
terminology in class discussions, in their writing, and on examinations.
This will ensure that the students understand the meaning of the words
and the appropriate times to use the words in a sentence. More specifically,
the council focuses on the words complementary and mutually exclusive
events. My seventh grade students can name the complement of an event,
however, my eighth graders have difficulty finding mutually exclusive events
and their probability. I also find it strange that these two terms are
grouped together when there is not a clear relationship between the two,
except that they both do not share any of the same components. Also, in
Montgomery County, these two terms are not in the same curriculum. Nonetheless,
I agree with the importance of the students learning the vocabulary and
using the terms regularly.
The next goal in the Probability objective is, "use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations" (NCTM, 2000, p. 248). The council suggests that the students make predictions before conducting the experiment. This will allow the students to test their conjectures about the experiment, which might correct misconceptions. This also may make the students consider other factors in the data they may not have realized were there before the data was collected. The more practice the students have at this skill, the more they will catch mistakes before getting a wrong answer.
The last objective, "compute probabilities
for simple compound events, using such methods as organized lists, tree
diagrams, and area models’ (NCTM, 2000, p. 248) is
a reasonable request for most students to be able to accomplish. However,
I do not agree with the level at which the standards document says the
students should construct tree diagrams. The writers have tied in the data
from the paper airplane experiment in order to ask questions, make predictions,
and compute probabilities. Many students have difficulty correctly constructing
organized lists and tree diagrams. If the students cannot conceptualize
what the question is asking, a diagram would be very difficult to create,
especially at the level of complexity the council is suggesting.
The Achieve document, Foundations
for Success: Mathematics for the Middle Grades, provides not only objectives
for teachers to follow, also included are sample problems and solutions
that teachers can use in class that students can complete for greater understanding
of the topic. After examining these sample problems, I have very little
confidence that they will enable most students in the middle grades will
gain an understanding of the topics discussed.
The Data strand is broken into
three sub categories. The first category discussed is Measurement and Approximation.
The objectives MAP says that students should be able to perform are very
broad. The range under this category spans from using measuring tools to
figuring out weighted averages to find unit rates. That does not even include
the other two subsections of the unit. Nevertheless, the goals can be achievable
in the middle grades. However, this document provides more then objectives
to obey, the sample problems also must be completed. The second problem,
How Big is the Cube?, given is unattainable by the average student and
incredibly difficult for a gifted student to decipher in a class period.
I think the problem is a terrific question and connects percents and measurement
in a way that would bring clarity to all students. However, the developmentally
appropriateness of the problem is not suitable for a middle school student.
Important changes in the thought processes are apparent in the emergence
and development of formal operations: from eleven years into adolescence
(Richmond, 1970, pg. 54). A student at the high
school level would be inclined to undertake such a math problem.
The objectives set forth in the
Data Analysis section are significantly more difficult then those in NCTM.
In this document, MAP lists what they believe students should be able to
understand as pre-requisite knowledge and what students should be able
to do in terms of the objectives. The pre-requisites are practically a
summation of the objectives NCTM gives for the same grade band. The actual
objectives are way out of the league of middle school students, with the
exception of constructing graphs and interpreting measures of central tendency.
Surprisingly, the sample problems are not as advanced as the objectives
suggest. I like the 50-Meter Dash problem; it incorporates histograms,
frequency tables, and broken line graphs. The connections are useful in
allowing the students to see the graphs relate to each other. Another appropriate
question is number eight, Math Scores. I have found that students have
difficulty noticing that the five numbers in a box plot is a percentage
of the range of scores. This problem explores measure of center and the
percentage of the data they each represent.
The objectives in the Probability
section are written in such a way that they seem to be appropriate for
students in grades 6-8. They also correlate with the NCTM standards of
comparing the probabilities of two or more events and conducting experiments.
The tenth sample problem, Flipping a Penny, is a problem I have solved
with my students with a high rate of success. The next problem, Rolling
a Dice, is a problem in the honors seventh grade math curriculum in Montgomery
County Public Schools. It is challenging for the students because it reviews
skills, such as multiples and sums. All in all, the problems in this section
are well thought out and provide a challenge for middle school students.
The Principle and Standards
document is a revision of the three previous documents standards documents-
Curriculum, Professional, and Assessment. The intention of these documents
is to be the principle vision of school mathematics. Although most school
districts use the standards as a guide in writing curriculums, there are
many opponents of the document. While I am not completely against the document,
I know that it is not entirely appropriate for the students in the levels
they suggest. In a letter to the editor of Mathematically Correct, one
critic agrees that the expectations of the standards are inappropriate,
It is not a set of standards in the usual meaning of the term, for it refuses
to say what exactly all children should learn in thirteen years of schooling
(Raimi, 2000, para.4).
Why do the documents insist that all students can accomplish all the goals presented? The answer is due to the results of the TIMSS; they feel that if other countries are doing it, so can we. This statement is absolutely false for many reasons. Reformers often compare the educational system in this country with that of Japan. The reality is, it is difficult to compare factors that may skew the results of the tests. Too often, people look at such findings as students’ achievements or particular instructional practices in isolation and jump to inappropriate conclusions or make inaccurate inferences. Cross-cultural comparison is a very difficult task, requiring good understanding of each other’s society (Watanabe, 2000, p.34). In most countries, the students tested are born and raised in that country. They live by the culture and speak the native language. The United States, on the other hand, is melting pot. Many of the students that live in this country and attend our schools where not born here and speak English as a second language. In this country, the student demographics of the percentage enrolled in public schools, grades K-12 are as follows: 17% African American; 3.9% Asian; 14.4% Latino; 1.2% Native American; 63.3% White. (Educational Trust, 2001). Essentially, a little more than one-third of the students in this country are were born in another country. Therefore, it is impossible to get accurate results when comparing this country to another, when some of the students here could be natives of the country being measured. The school I teach in has an ethnic breakdown representative of that of the United States: 15.4% African American; 6.6% Asian; 16% Latino; 0.2% Native American; 61.8% White (Montgomery Country Public Schools, 2001). My schools results of the countywide Math Criterion Reference Test (CRT) of those sixth graders who met the standards are as follows: 29% African American; 33% Asian; 29% Latino; 71% White. (MCPS, 2001). Clearly, the white students performed exceedingly higher than their peers Certainly, some of the non-white students could be from this country, know the culture, and speak English as their primary language. However, according to the data above, if students that are not citizens of this country were exempt from the test, the results would surpass the current standings. This would show that we would be superior to countries that currently hold a higher ranking. In which case the teachers and the students would not be under such scrutiny to live up to higher expectations.
Comparisons such as these can result in a correlation between mathematics achievement and attitudes towards mathematics. Students can develop math anxiety from teachers, parents, and society’s attitude toward math. Teachers often feel pressure from their school district or state to make sure their students perform well on assessments. Unfortunately, this can result in the teacher teaching what they think will be on the test and drilling it into their students brains so they can regurgitate what they learned. Invariably, the students will forget what was ever taught to them as soon as the test is completed. A bad experience in a math class or with a math teacher can cause math anxiety. Evidence suggests that math anxiety results more from the way subject matter is presented than from the subject matter itself (Greenwood, 1984, pg. 662). Teachers that sit at their desks and talk at the students or only go over problems will cause their students become intimidated, causing them not to ask questions, causing them to fall behind. This will result from the [student’s] inability to handle frustration, and emphasis on mathematics through drill without understanding (Fiore, 1999, p. 403). When teachers are feeling confined by what they are to teach, they could lose interest and it will show in their teaching. If students think that the instructor is not happy teaching…they will be less motivated to learn (Jackson & Leffingwell, 1999, p. 585). The demands these documents are putting on the school system are trickling down into the classroom and having negative effects on the students.
The two documents Principles and Standards and Foundations for Success are great full of ideas for improving mathematics curriculum. Not all of their ideas are appropriate for the grade level they suggest. The former president of NCTM states, "They are a statement of where we want to be. None of us will agree with every word, but we don’t have to, especially on details of implementation" (Lappan, 2001, para. 3). Both documents do not make any references to lending the teachers any freedom to make professional decisions about what should be taught at what levels.
Moreover, for all students to accomplish these goals is unreasonable. Unfortunately, what began as a noble process to help low-income children achieve at higher levels has become an educational albatross that punishes both teachers and students and declares that schools are ineffective when all children do not learn at arbitrary level predetermined by individual external to the school (Thomas & Bainbridge, 2001, p. 662). This is very true. Test scores only show the results and does not discuss the factors behind the results. For example schools that have a high minority rate, or schools that have low funding. Low achieving schools (based on test scores) are put on probation and supervised closer by the superintendent. But no body ever looks to see the improvements that may have been made, based on previous tests, and improvements that continue to be made.
I could not describe the students in the schools in this country in one word. All of the students are very diverse. Whether it is their family’s background, learning styles, developmental stage, or achievement level. All students in this country are different, and these two organizations-National Council of Teachers of Mathematics and Achieve does not recognize the discrepancies in the data. They focus on improving numbers when compared to other countries, without realizing the infrastructure of the different countries around the world.
The objectives set forth by NCTM
and Achieve are mostly unattainable for all students to accomplish due
to the level of difficulty. These organizations believe all students can
met all of the expectations regardless of external factors. They need to
reexamine the statement that all students can learn the same objectives
at the same level of schooling. Their demands are completely unrealistic.
Sure, some students at the middle school level can accomplish the goals.
Some students, however, need special education services, and some find
math to be very challenging. When students feel as though they are being
rushed through a topic they do not understand, they will get anxious and
will not be able to perform as well as they normally would. It is acceptable
to have high expectations for students, however in order to produce success,
the goals need to be within reach of the student’s ability.
Fiore, G.
(1999). Math-Abused Students: Are We Prepared to Teach Them?
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Jackson, C & Leffingwell, R. (1999). The Role of Instructors in Creating Math Anxiety in Students from Kindergarten through College. Mathematics Teacher, 92 (7), 583-586.
Lappan, G. (n.d.) We Have our Principles and Standards: What now? Retrieved December 12, 2001 from http://www.nctm.org/news/pastpresident/2000-04president.htm
Mathematics Achievement Partnership (2001). Foundations for Success: Mathematics for the Middle Grades, Achieve, Inc.
Montgomery County Public Schools (2001). Monthly Report of Enrollment by Grade, Gender, and Race.
Montgomery County Public Schools (2001). Number Tested and Percent Met Standard in CRT Math (Math A, B, C).
National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: Author.
Raimi, R. (2000, October 20). Standards in School Mathematics. Retrieved December 12, 2000 from http://www.mathematicallycorrect.com
Richmond, P.G. (1970). Introduction to Piaget. New York: Basic Books, Inc.
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