**Proposal for a Hands-On Math Lab**

**May 4, 1995**

**Fran Endicott Armstrong, Ph.D.**

**Mathematics Department**

**St. Louis Community College at Meramec**

**11333 Big Bend Blvd.**

**St. Louis, MO 63122-5799**

**(314) 984-7769**

**armstl@earthlink.net**

**Features
of a Hands-On Math Lab**

**The
Hands-On Math Lab at SLCC/Meramec**

**Three Sources of the Problem of Poor
Learning of School Mathematics**

**I believe that the failure of many students of normal intellectual
ability to learn school mathematics stems primarily from three situations
prevalent in mathematics education:**

**Many students try to do mathematics by obedience, attempting to learn and remember what rules to apply in what situations;****Students are forced to move in lock-step with their usually age-matched and later ability-matched math class of usually 20 or more students;****In mathematics classes students are not routinely expected to think and solve novel problems that are genuine "problems" for the individual in that the student does not immediately have available an algorithm for solving.**

**1.) Many students try to do mathematics by obedience.**

*** Many students do mathematics by obeying rules and/or following
procedures for doing computational exercises in arithmetic and algebra. For example,
they memorize the rule that to add or subtract fractions they need to rewrite
the fractions with a common denominator and then add or subtract the numerator
and keep the common denominator.**

*** The problem is that for too many students these rules in arithmetic and
algebra have no meaning. The rules just don't make sense to some students
because they lack an experiential understanding of the process. As a result,
the students have difficulty remembering the rules and, in particular, when to
apply the rule. For example, there are students who, when asked to multiply
fractions, will first determine a common denominator. This is not incorrect,
but rather it is unnecessary and inefficient.**

*** Furthermore, students who do not understand the rules and procedures do
not comprehend the relationships among rules. For example, some students do not
realize that the same reason underlies the following:**

· **Using a common denominator when
adding and subtracting fractions.**

· **Lining up the decimal points when
adding and subtracting decimal numbers.**

· **Combining like terms when adding and
subtracting polynomials (e.g., 3x y - 2x + 2x y + 4 = 5x y -
2x + 4)**

**Therefore, these students have a plethora of isolated, meaningless rules
to try to remember and apply.**

**Other Difficulties related to doing
Mathematics by Obedience**

**There are other phenomena related to the learning of mathematics as
isolated rules without meaning. Many students do not realize that there may be
more than one way to perform a computation or solve a word problem. And many
students who can perform standard arithmetic computational algorithms are
uncertain which arithmetic operation(s) to perform to solve some single-stage
and especially multi-stage arithmetic word problems. These difficulties are
seen not only in students who demonstrate generally low academic ability across
subjects but also in students of average and above-average intellectual
ability. Furthermore, these difficulties often result in math phobia and test
phobia particularly as these students attempt to move on through algebra.**

**2.) Moving classes in lock-step progression in mathematics lessons leaves
some students overwhelmed and others bored.**

**Another problem with school mathematics is that students are required to move
in lock-step with their class. There are occasionally some attempts to allow an
individual or a small group to work separately from the rest of the class.
Usually, the more advanced students are allowed to work in the textbook of the
next grade level while the teacher works with the regular students or the
slower students. But, in general, students are moving in groups through
textbooks. The teacher teaches the same lesson to the group and all the
students in the group do the same in-class written assignment and/or the same
homework. Whether the classes are grouped homogeneously or heterogeneously, the
individuals in the group are all required to be taught the same lesson and do
the same exercises.**

**Grouping students by age in elementary school is a convenience and an
economy of the system. One adult can then teach/manage a group of about 15 to
30 students. This grouping of students is more problematic in mathematics
than in other subject areas due to the cumulative nature of mathematics. It
really is rather difficult to learn arithmetic operations on fractions and/or
decimals without first understanding operations with whole numbers.**

**Most teachers really know that moving classes in lock-step through math
lessons doesn't work. The class moves too slowly for the students who pick up
mathematics quickly and too fast for the students who are slower at
comprehending mathematics. Students who understand mathematics quickly are
bored at being told the rules rather than getting to figure them out for
themselves, and they are frustrated waiting for the rest of the class to catch
on. And other students are generally confused and unsure of why they're doing
what they are told to do in math class. They try to follow the rules but
usually don't understand why the rules work and sometimes don't even quite
follow what it is the teacher wants them to do. Many times, when these students
are just about to catch on, the class moves on again.**

**And this frustrating cycle is repeated year after year. Many years in
grade school, the textbook begins with place value and adding whole numbers and
then subtracting, and then on to multiplying and dividing (with long division
using all four arithmetic operations in some apparently incomprehensible system
of computation and checking) and then on to decimals and fractions and then, in
later years, on to ratios, proportions and percents. There may be some geometry
and probability and statistics, but these more intriguing topics are usually in
the chapters at the end of the book which are rarely covered. And each year,
some students are bored and some are confused and overwhelmed; hardly any
student is being taught at the level where s/he is.**

**Some students come to identify mathematics with the trivialities of
arithmetic as taught in school and do not have the opportunity to see the
beauty of the larger picture of mathematics or even taste the delight of
playing with numbers (although some students do, on their own, entertain
themselves with noticing interesting things about numbers and operations). And
other students become accustomed to not understanding what's going on in math
class. Some eventually settle for doing the minimum to get by (grade-wise)
in math class and others "work their buns off" trying to keep all
those meaningless rules straight and remember which rules the teacher wants
them to apply in particular word problem situations.**

**3.) Most students do not get enough experience solving problems since
what constitutes a "problem" is quite individual.**

**Many students come to believe that one is not expected to think in math
class. Some students find the "problems" presented in math class to
be trivial. Others struggle to cling to magic words or phrases as clues to what
operation(s) to use. The difficulty is that what constitutes a problem is very
specific to an individual at a particular time. A true problem is a
situation in which an individual does not immediately have available an
algorithm to get to the solution. The word problems provided in most
textbooks, even those word problems with extraneous information not needed for
solution and those with insufficient information, are trivial for the very
bright students and overwhelming for many others.**

**Each student ought to be able to encounter problems in mathematics that are
at an appropriate level of difficulty for that individual at that time so that
s/he has the opportunity to be in a situation in which s/he truly has to think,
try various strategies and most often eventually succeed in solving, possibly
with the benefit of discussing the problem with other students who are at about
the same problem-solving level. This way, each student would have the
opportunity to struggle with a problem, to learn to persevere and try various
methods in solving, and to experience success in problem solving. Also, it is
important for students to have the opportunity to explain their thinking to
others and also to see other students' methods of solution. Obviously, problems
need to be tailored to the individual and individual students need to be
allowed to discuss problems with whatever students are working on the same
problem, regardless of age or grade level. And the grade level of a problem
should be nonexistent.**

**The solution: Mathematics by understanding instead of by obedience
results in more effective learning.**

**A better way of doing and learning mathematics, and one which produces
more effective and lasting knowledge, is mathematics by understanding. But this
requires relevant experiences on the part of the learner that lead to his/her
understanding of numbers and operations on them. We humans learn initially
through our senses. The more sensory avenues through which experiences with
materials can impact the individual, the more likely that relevant learning
will occur.**

**An Important Clarification**

**I am not blaming teachers for these problems of students being forced to
move in lock-step through mathematics lessons that students eventually come to
see as a bunch of rules to be obeyed along with a dearth of problem solving
tailored to the individual student. Many teachers strive to teach mathematics
meaningfully and try to keep the slower students from being overwhelmed. Most
teachers would like to provide ways to keep the students who catch on quickly
engaged in stimulating mathematical endeavors and provide relevant problems for
all. But the structure in which teachers effort to do all these things makes
this all but impossible. Fifteen or more students in one room matched by age
but of diverse abilities in mathematics --let's be real, what can a teacher do
to satisfy each students' needs in learning mathematics?**

**Question: What can be done to help students learn arithmetic and move on
to meaningful learning of algebra?**

**Answer: Enlightened utilization of a Hands-On Mathematics Laboratory.**

**Essential features of a Hands-On Math Lab include the following:**

**1)Manipulative materials and accompanying guided discovery worksheet
sequences**

**The heart of the successful math lab is appropriate mathematics
manipulative materials and accompanying worksheet sequences of guided discovery
lessons and problem solving situations. Examples of relevant and versatile
manipulatives include Cuisenaire Rods®, pegboards, base ten blocks, Pattern Blocks,
Eli's Magic and Regular Peanuts, Henry Borenson's HANDS-ON EQUATIONS®, Henri
Picciotto's ALGEBRA LAB GEAR®.**

**2)Self-paced progress with students working individually and/or in small
groups.**

**Each student is encouraged to work at his/her own pace with the materials
individually or in pairs or in small groups. These groupings are self-selective
and fluid in the sense that they can change as students move at different paces
and/or individuals find certain classmates more or less compatible.**

**3) Backup pencil and paper exercises.**

**These exercises can be done by the student outside of the lab at
appropriate times after the relevant guided discovery interaction with the math
manipulatives.**

**4) Problems (as opposed to exercises) and exploratory investigations.**

**Problems and exploratory investigations should be provided that require
the student(s) to think and to analyze and to decide which previously-learned
mathematical knowledge and skills to apply in the situation. The individual may
also learn new concepts and principles through efforts to solve the problem.**

**5) Math-related tools, both low-tech and high-tech.**

**Opportunities need to exist for students to learn how to use appropriate
math-related tools such as rulers, compasses, protractors, arithmetic calculators,
scientific calculators, graphics calculators, data-collecting devices (e.g.,
Texas Instruments CBL) and computer software as tools to employ in mathematical
problem-solving.**

**6) Caring and competent mentors**

**An essential component of the math lab is caring and competent mentors
who are knowledgeable about the use of the materials in the math lab and
methods of guiding students in using the materials and learning problem-solving
strategies. Appropriate training of the mentors (probably in a hands-on math
lab situation) is crucial.**

**Question: Why not have each teacher (or those who want to do so) set up a
Math Lab area in the classroom rather than having a Math Lab for the whole
school (or one in each wing or floor of a large school)?**

**Answer: Besides the waste of resources in the replication of the
manipulative sets in each classroom and the probably inadequate space in many
classrooms to set up a Math Lab Area, there are more important reasons why
trying to implement a Math Lab in individual classrooms is contraindicated. Let
me assure you from personal experience that, even with a modest class size of
from 16 to 20 elementary school students, the effort to respond to the
students' legitimate needs for assistance and attention during class and the effort
to keep up with the paper work outside of class is well beyond the capacity of
an individual teacher. The Math Lab situation is labor-intensive in the
guidance and assistance of students as well as in the checking of papers and
monitoring of students' activity and progress. Additional staff beyond the
classroom teacher are absolutely necessary. Some of these trained Math Lab
staff may be paid professionals and others volunteer parents and/or neighbors
and/or retired persons.**

**Areas to Consider in Developing a Hands-On
Math Lab**

**1) Physical space and furnishings**

· **tables and
chairs (preferably not individual desks for most of the students)**

· **a small room
or alcove and possibly study carrels for people who need isolation from
distraction and also for testing**

· **cabinets for
securely storing the manipulatives**

· **file cabinets
or boxes for the folders of worksheets**

· **file cabinets
or boxes for the students' file folders**

· **chalkboards
and/or dry-erase boards**

· **overhead
projector and screen**

· **a place and
equipment for viewing videotapes**

· **bulletin
boards**

· **informative
and/or inspiring decorations (There are excellent math posters commercially
available. Student-produced posters would be useful as well.)**

**2) Manipulatives and related worksheets**

**Manipulatives should be versatile, that is, useful for modeling several
important concepts. An excellent example of a versatile math manipulative set
is Cuisenaire Rods which can be used to develop concepts and operations
relative to natural numbers, fractions, perimeter and area. Furthermore,
Cuisenaire Rods lead to the use of base ten blocks and also Algebra Lab Gear.
Excellent books of black-line masters are commercially available and can be
useful in guiding students' use of Cuisenaire Rods in developing these concepts
of number and operations and geometry as well as presenting problem-solving
situations.**

**3) Relevant Tools (both low tech and high tech)**

**Some examples include the following: rulers (with a progression from
rulers marked only in centimeters to those marked in centimeters and
millimeters and/or inches to sixteenths of an inch), tape measures, compasses,
protractors, counting beads, abaci of various sorts, templates of shapes, etc.**

**Electronic tools include arithmetic calculators, scientific calculators,
graphics calculators, data-gathering devices (e.g., Texas Instruments' CBL),
computers and problem-solving and/or exploration software, etc.**

**4) Staffing (trained professionals and trained volunteers) and their
Training.**

**The Hands-On Math Lab should be headed by a Coordinator and, if possible,
an Assistant (or Associate) Coordinator, one of whom should probably be active in
the Lab whenever the Lab is open. An Associate Coordinator would be especially
useful if the Lab is used not only during the school day but also in the
evenings (Please see Scheduling below).**

**The classroom teachers who bring their classes also need training since
they will be involved in guiding their students while in the Lab and also
evaluating students' learning.**

**Math Lab Mentors may be needed in addition to the Coordinator and the
classroom teacher. These Mentors need to be familiar with all aspects of the
Lab and assist the classroom teachers with guiding students' learning in the
Lab and checking papers.**

**Educational assistants may be trained to present specific lessons with a
particular manipulative to individuals and/or small groups as needed. The educational
assistants can also check students' papers and keep materials in order as well
as restock files of worksheets.**

**For a modest class size of about 20 to 24 students, probably the minimum
staffing needed would be the Math Lab Coordinator the classroom teacher and a
trained volunteer on duty. The volunteer could assist with checking papers and
showing students how to use some materials. The classroom teacher and the
Coordinator would also need to share in the work of checking students' papers
in this minimum staffing situation.**

**Excellent training for staff for a new Hands-On Math Lab would be
available through the courses Hands-On Arithmetic and Hands-On Preparation for
Algebra at St. Louis Community College at Meramec starting in 1996 or through
Math Labs in operation in elementary or middle schools. Classroom teachers, the
Math Lab Coordinator and the Mentors need to become familiar with the
manipulatives and related guided discovery, exploration and problem solving
worksheets, methods of guiding students in using the materials to discover
concepts and principles, and implementation of a Math Lab including developing
methods of monitoring and evaluating students' progress.**

**5) Expectations regarding appropriate behavior in the Math Lab**

**There needs to be agreement about this among the staff (including the
classroom teachers) and these expectations elicited from and/or communicated to
students in all classes.**

**6) Scheduling of the use of the Hands-On Math Lab**

**Classes should use the Lab for regular daily math instruction possible
two to four times a week most weeks throughout the school year. There is the
obvious need to avoid overcrowding as well as under-utilization of the Lab.**

**In many schools, the Hands-On Math Lab can be used evenings and weekends
as well as during the school day. After school and evening and weekend sessions
can be used by the following:**

· **daytime
students for further progress; Extended day programs can allow students to go
to the Math Lab staffed after school for math-related games which provide
further math stimulation and practice and strategy development.**

· **training of
teachers; Math Lab Mentors and educational assistants.**

· **adult basic
education.**

**7) Methods of Monitoring students' Activities and Progress in the Lab**

**A reasonable and dependable method must be developed for recording what
materials a student used and what worksheets and/or explorations or problems a
student did on which days in the Lab. It is probably best for most students not
to work day after day with the same materials. Some spiraling of lessons is
useful both to prevent boredom and also to keep the student moving forward on
several fronts (e.g., maybe exploring "Fractions with Pattern
Blocks," using the pegboard to develop concepts of prime and composite
natural numbers, and using "Hands-On Equations" to drill arithmetic
at the same time as solving algebraic equations).**

**students' worksheet and problem solutions need to be checked. If there
are errors, the student needs to be directed to a Mentor for assistance and
clarification. Some worksheets and problem solutions need to be selected for a
portfolio of a students' work. And the students' progress and learning needs to
be recorded and evaluated for reports to parents.**

**The Self-Paced Hands-On Math Lab at St. Louis
Community College at Meramec**

**The mathematics department at St. Louis Community College at Meramec
offers three developmental courses -- Basic Mathematics, Elementary Algebra and
Intermediate Algebra. Basic Math is roughly grade-school arithmetic and
intuitive and measurement geometry. Elementary Algebra is about the same
content as 9th-grade Algebra I. However, Algebra I in junior or senior high
school is normally taught in about 160 to 180 class hours over about nine
months of the school year whereas, at the community college, Elementary Algebra
is most often taught in three 50-minute classes per week for 16 weeks or two
80-minute classes per week for 16 weeks. This amounts to about one-third the in-class
time compared with the 160 50-minute class periods of Algebra I in high school.
Students enrolled in Elementary Algebra at the community college usually fall
into one of these three categories: (1) they have never had algebra before in
their lives, (2) they have taken an algebra course or two but did not learn it
well enough to demonstrate adequate competence in algebra on the placement test
or (3) they took an algebra course some time ago and have forgotten much of it.
Is it a wonder that students in the first two categories in particular often do
not succeed in our community college Elementary Algebra course taught in
one-third the student-teacher contact time.**

**Furthermore, many of our community college students in Basic Math and
Elementary Algebra primarily attempt to do mathematics by obedience. And for
many of them, doing mathematics by obedience doesn't work in the long run.
Often, by about the middle of Elementary Algebra, there are just too many rules
for some students to keep straight in their heads and also know when to apply
which rule. This is because, for some of these students, the rules are
literally meaningless.**

**After failing to succeed in Elementary Algebra (that is, getting a D, or
W for withdraw or an F) for one or several semesters, some students are
desperate enough to be willing to try another approach to learning math. Due to
experiences in math classes prior to college, some students are so frightened
of mathematics that they put off enrolling in math classes at the college for
several semesters. These students might even be willing to try an alternative
approach to learning math before daring to enroll in a regular math class at
the college. For those students who do poorly in the developmental classes
because they are still trying to learn math without bothering to do the written
assignments, I have no solution except to wait for them to mature and become
motivated.**

**To provide an opportunity for our adult students who are willing to do
the work in math but are afraid of math or who may have had poor experiences in
previous math classes and do not really understand arithmetic and/or algebra, I
have developed two courses intended to be taught in a self-paced situation in a
Hands-On Math Lab involving significant interaction with relevant math
manipulatives. The courses are Hands-On Arithmetic and Hands-On Preparation for
Algebra. But the Hands-On Math Lab is not only for students who need help with
preparing for developmental courses. In the Hands-On Math Lab we can also
provide some hands-on experiences on topics such as infinite sequences and
series, conic sections, the binomial expansion which are relevant to higher
level math classes. Likewise, geometry and probability and statistics topics
could also be featured in the Hands-On Math Lab.**

**Another clientele that may find these courses in our Hands-On Math Lab
useful are teachers of mathematics at all levels from elementary school through
high school and teachers of adult education. Teachers of the gifted as well as
teachers of the learning disabled and also teachers of the blind and teachers
of the deaf would find these courses useful. College students who are planning
to go into the teaching profession may want to take these courses to help them
with their college math courses as well as to prepare for their teaching
careers. Teachers who plan to develop a Hands-On Math Lab in their school can
experience for themselves such a learning situation by taking these courses.
Parents who home-school their children or who just want to provide enriched
mathematics experiences outside of school for their children will find these
courses beneficial. People who want to do individual math tutoring will profit
from learning about these materials and methods in the Hands-On Math Lab.**

**Each of the courses, Hands-On Arithmetic and Hands-On Preparation for
Algebra, are offered as 5-credit-hour courses available both in the daytime and
in the evening. Students from both courses can be working in the Hands-On Math
Lab at the same time. In the fall or spring semester, students would be
expected to spend a minimum of 15 hours a week on their work for the
course, much of that time in the Hands-On Math Lab. Since the 8-week summer
session is half as long as the fall or spring semester, students taking a
Hands-On Math Lab course in the summer session would expect to spend a minimum
of 30 hours a week on the course.**

**MTH:513 Hands-On Arithmetic**

**5 credit hours**

**Course description:**

**Working individually and/or in small groups, on worksheets and problems
students will use various mathematics manipulatives (Cuisenaire Rods,
pegboards, pattern blocks, base ten blocks, magic and regular peanuts,
first-quadrant rectangular coordinate graphs, etc.) to explore mathematical
concepts in order to gain understanding of numbers (natural numbers, whole
numbers, fractions, decimals, integers and percents) and arithmetic operations
on them as well as metric measurement. This course is designed especially for
students who want to acquire an understanding of arithmetic rather than merely
acquiring rote learning of the rules for arithmetic operations. This course is
also valuable for teachers (including parents and tutors) who want to be
effective in helping students understand arithmetic. This course does not
replace Basic Mathematics, but it can provide a thorough foundation that may
enhance the probability for successful learning in Basic Mathematics. In this
Hands-On Math Lab course, students proceed at their own pace and, if
progressing, may re-enroll for subsequent semesters as needed. Students who
proceed more quickly may complete the course work in less than a semester.**

**MTH:514 Hands-On Preparation for Algebra**

**5 credit hours**

**Course description:**

**Working individually and/or in small groups on worksheets and problems,
students will use various mathematics manipulatives (Dr. Henry Borenson's
Hands-On Equations®, Henry Picciotto's Algebra Lab Gear®, geoboards,
four-quadrant rectangular coordinate graphs, etc.) to explore algebra concepts
in order to gain understanding of solving equations and inequalities,
performing operations on polynomials and graphing in the rectangular coordinate
plane. This course is designed especially for students who want to understand
algebra rather than merely acquiring rote learning of pencil and paper
algebraic manipulations. This course is also valuable for teachers (including
parents and tutors) who want to be effective in helping students understand algebra.
This course does not replace Elementary Algebra, but it can provide a thorough
foundation that may enhance the probability for successful learning in the
algebra sequence. In this Hands-On Math Lab course, students proceed at their
own pace and, if progressing, may re-enroll for subsequent semesters as needed.
Students who proceed more quickly may complete the course work in less than a
semester.**

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11333 Big Bend Boulevard, St. Louis, MO 63122-5799

Phone: 314-984-7500 | TDD: 314-984-7704

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URL: http://www.stlcc.cc.mo.us/mc/

Updated 11/1/99