Proposal for a Hands-On Math Lab
May 4, 1995
Fran Endicott Armstrong, Ph.D.
St. Louis Community College at Meramec
11333 Big Bend Blvd.
St. Louis, MO 63122-5799
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:
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 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
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
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.
St. Louis Community College at Meramec
11333 Big Bend Boulevard, St. Louis, MO 63122-5799
Phone: 314-984-7500 | TDD: 314-984-7704
An Equal Opportunity/Affirmative Action Institution
Accommodations Available to Persons with Disabilities