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Proposal for a MathStrongLab

 

 

 

Proposal for

a

MathStrongLab

 

May 4, 1995

(revised July 7, 2008)

                                              

by

 

Fran Endicott Armstrong, Ph.D.

 

 

 

 

 


The Problem

 

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 contemporary school mathematics situations, as follows:

 

1.)         Many students try to do mathematics by obedience, attempting to learn and remember which rules to apply in which situations; 

 

2.)         In mathematics classes students are not routinely expected to think and solve novel problems that are genuine "problems" for the individual student.

 

3.)         Students are forced to move in lock-step with their usually age-matched, and later, ability-matched mathematics class;

 

I will also explain here how even if some of the popular contemporary curricula avoid the first two difficulties above, that they still fall prey to the trap of moving students in lock-step with their class.

 

Let's consider each of these.

 

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

 

Many students attempt to do mathematics by obeying rules and/or following procedures for doing computational exercises in arithmetic and algebra.  For example, they memorize this algorithmic procedure:  to add or subtract fractions, rewrite the fractions with a common denominator and then add or subtract the numerators and keep the common denominator.

 

The problem is that for too many students the rules in arithmetic and algebra have no meaning.  The rules don't make sense to these 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 which rule.  For example, there are students who, when asked to multiply fractions, will first determine a common denominator, misapplying the rule for adding fractions.  Using a common denominator when multiplying fractions 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:

 

          Use a common denominator when adding and subtracting fractions.

 

Line up the decimal points when adding and subtracting decimal numbers.

 

Combine like terms when adding and subtracting polynomials   (e.g., 3xy - 2x + 2xy + 4 = 5xy - 2x + 4)

 

These three rules can all be subsumed under the generic dictum that we add and subtract like things.

 

Therefore, these students have a plethora of isolated, meaningless rules to try to remember and apply rather than fewer but more broadly applicable rules.

 

 

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 low academic ability across subjects but also in students of average and even 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.)              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 mathematics class.  Some students find the "problems" presented in mathematics 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 or does not know immediately which of the available algorithms applies to this situation.  The word problems provided in most textbooks, even those word problems with extraneous information not needed for solution, are trivial for the mathematically-talented students and overwhelming for many others.  Each student ought to be able to encounter problems 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.  The grade level of a problem should be nonexistent.

 

In some of the contemporary curricula being adopted by many schools and districts, the emphasis is on activities intended to enable students to develop skill with numerical operations through various projects and games and to engage in solving of relevant problems.  However, the difficulty persists; what constitutes a problem is specific to an individual at a particular time.  So the students who already realize how to solve the intended “problem” often try to tell their classmates how to solve the problem and often the teacher expects them to do precisely that.  Hence, some other students who might have been able to solve the problem on their own are robbed of that opportunity.  And those students who were overwhelmed by the problem again are demeaned because they are aware that they are clueless while the “smart” kids knew how to do it and the other kids understood when the smart kids told them how to do it. 

 

Many of the testing programs used in the various States feature a section in the mathematics test on problem solving.  Students are encouraged to show and explain all their steps so as to receive full credit relative to the grading rubric.  And items are given to all the students in a particular grade that are intended to be genuine problems for all of the students in that grade.  But how can that possibly be the case?  Again, for some students those “problems” will be simply exercises since they know immediately what algorithm they have available to get the answer.  However, those students may lose points relative to the rubric since they don’t “show all the steps” because for those mathematically talented students who can intuit the solution, there are not so many steps.   Other students will simply be overwhelmed and not have a clue what to do.  And some students in the middle will have what is for them a genuine problem with which to deal and some of them will figure it out and show effectively how they solved the problem.

 

 

3.)              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 or through a curriculum.  The teacher teaches the same lesson to the group and all the students in the group do the same in-class activities and/or 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 at the same time.

 

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 is rather difficult to learn arithmetic operations on fractions and/or decimals without first understanding operations on whole numbers.  So, if a student doesn’t understand whole number operations, moving ahead with the group to fractions and/or decimals can prove problematic.

 

Most teachers realize that moving classes in lock-step through mathematics 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 being allowed 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 because they don't understand why the rules work, these students often don't quite follow what it is the teacher wants them to do.  Usually, when these students are just about to start to catch on, the class moves on.  And this frustrating cycle is repeated year after year.  Furthermore, in intermediate and/or middle grades, the textbook begins year after year with place value and adding whole numbers and then subtracting, and then on to multiplying and dividing whole numbers (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 rarely covered.  And each year, some students are bored and others 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.  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 mathematics class.  Some uninspired students eventually settle for doing the minimum to get by (grade-wise) in mathematics class.  The diligent, dutiful students work very hard trying to keep all those apparently meaningless rules straight and remember which rules the teacher wants them to apply in particular word problem situations.

 

Some of the contemporary curricula being adopted by many schools and districts do not challenge the status quo of the organization of mathematics classes.  The delightful activities, games, and problems featured in the curriculum are still delivered to the class in lock-step.  Teachers are directed not to be concerned about those students who don’t quite “get it” on the first round but rather to have faith in the curriculum and stay on schedule.  Due to the spiral nature of the curriculum, the curriculum vendors explain, those students will have several opportunities in the future to “catch on.”  This may be true from solely a cognitive point of view.  However, this way of thinking disregards the students’ affect.  One thing that students who have difficulty with mathematics understand very well is that they don’t get it and the other students do.  This awareness smashes their self-concept with respect to learning mathematics.  Such students usually come to at least one of the two following conclusions:  (1.)  I can’t learn mathematics, and/or (2.)  Mathematics is meaningless and useless.  And so they ask, at least inside themselves, “Why should I have to learn mathematics anyway?”  Even ‘though these students are physically present in mathematics class, very often they have checked out mentally.  Their negative affect with respect to mathematics so paralyzes them mentally that they are unable to cognitively engage in the mathematical activities.  Too many students continue to suffer in mathematics class.

 

 

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 learning 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 avenues through which experiences can impact the individual, the more likely that relevant learning will occur.

 

Students can only truly learn by understanding if each student is allowed to progress at his/her own pace.

 

If students are moved in lock-step through the curriculum, the “slow” students will often just look over at a “smart” kids paper and copy.  Or the teacher or the “smart students” will often tell the slower students the strategy to use in a game or how to apply an algorithm or how to solve the problem thus stealing from them the opportunity to figure things out for themselves.   And still the very mathematically talented students are also limited by being held back to what the class is doing rather than being allowed to progress the way they could.

 

In order to allow students to move at their own pace using materials and methods that enhance the likelihood of learning mathematics through understanding, Mentors are needed, at a preferred Mentor-to-Student Ratio, who are competent in using the materials and methodology of a self-paced hands-on math lab—a MathStrongLab.

 

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 student’s needs in learning mathematics?

 

 

A MathStrongLab

 

Essential features of a MathStrongLab 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, Henry Borenson's Hands-On Equations® Learning System, 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 which require the student(s) to think and 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.)        Mathematics-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, graphing calculators, computer algebra systems, and data-collecting devices (e.g., Texas Instruments CBL) as tools to employ in mathematical problem-solving.

 

6.)        Caring and competent mentors

 

An essential component of the MathStrongLab is a cohort of 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, experiencing for themselves learning in a MathStrongLab, is crucial.

 

 

Question:  Why not have each teacher (or those who want to do so) set up a MathStrongLab area in the classroom rather than having a MathStrongLab 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 MathStrongLab, there are more important reasons why trying to implement this 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 MathStrongLab situation is labor-intensive in the mentoring of students before worksheets, checking worksheets and providing individual recitation by students as part of checking, and monitoring of students' activity and progress.  It is absolutely necessary to have an adequate number of staff in addition to the classroom teacher.  Some of these trained MathStrongLab staff may be paid professionals and others may include volunteer parents and/or neighbors and/or retired persons.

 

 

Developing a MathStrongLab

 

Areas to Consider in Developing a MathStrongLab

 

1.)              Physical space and furnishings

 

·        tables or desks and chairs

 

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

 

·        shelves or secure cabinets for storing the manipulatives

 

·        file cabinets for the file folders of guided discovery worksheets

 

·        file cabinets for the students' file folders of their completed    worksheets

 

·        a board on which mentors and/or students can write

 

·        bulletin boards for posters, displays, and information

·        a place and equipment for viewing audiovisual material

 

·        informative and/or inspiring decorations whether commercially produced or supplied by the mentors and/or students

 

2.)              Manipulatives and related worksheets

 

Manipulatives should be versatile, that is, useful for modeling several important concepts.  An excellent example of a versatile mathematics manipulative is Cuisenaire® Rods which can be used to develop concepts and operations relative to natural numbers and fractions as well as geometric concepts of perimeter, area, volume, and surface area.  Furthermore, Cuisenaire® Rods lead to the use of Base Ten Blocks and also Algebra Lab Gear®.  Excellent books of blackline masters are commercially available and can be useful in guiding students' use of Cuisenaire® Rods in developing concepts of number and operations and geometry as well as presenting useful problem-solving situations. 

 

Pegs and Pegboards are an excellent complement to Cuisenaire® Rods since they represent number and operations in a discrete way (since the holes on the pegboards are separated from each other) while Cuisenaire® Rods can be used to model the same concepts in a continuous way (since each rod is a continuous piece).  Modeling the same concepts in these two fundamentally different ways, discrete and continuous, results in a more robust abstraction made by the student.  Therefore these concepts and rules discovered by the student will be more versatile in applying to diverse situations.

 

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

 

These tools include the following: rulers (with a progression from rulers marked only in centimetres to those marked in centimetres and millimetres and/or inches to sixteenths of an inch), tape measures, compasses, protractors, counting beads, abaci of various sorts, templates of shapes (e.g., Pattern Blocks, Tangrams, Pentominoes, etc.).

 

Electronic tools include arithmetic calculators, scientific calculators, graphing calculators, data-gathering devices (e.g., Texas Instruments' CBL), computer algebra systems, etc.

 

4.)              Staffing, both professionals and volunteers, provided with appropriate and adequate professional development in a MathStrongLab

 

The MathStrongLab should be headed by a Coordinator and, if possible, an Assistant (or Associate) Coordinator, one of whom should 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 after school and in the evenings (Please see Scheduling below).

 

The classroom teachers also need the appropriate professional development in the MathStrongLab since they will also serve as Mentors guiding their students while in the Lab and also evaluating students' learning.

 

The Coordinator and the Classroom Teacher will serve as MathStrongLab Mentors, but additional Mentors will be needed.  The Mentors need to be familiar with all aspects of the Lab and trained to implement the MathStrongLab coaching strategies to guide students' learning.

 

The MathStrongLab may also have, in addition to Mentors, paid or volunteer Mentor Assistants who may be trained to help individuals and/or small groups of students with just one or two particular manipulatives (e.g., using Pattern Blocks for fraction concepts) as needed.  The Mentor Assistants can also keep materials in order as well as restock files of worksheets.  As a Mentor Assistant receives further training and expands his/her repertoire of math manipulatives and knowledge of MathStrongLab guidance strategies, s/he may progress to becoming a Mentor.  In general, I recommend a Mentor-to-Student ratio of one to five.

 

As MathStrongLabs come into operation in elementary or middle schools, they can also be used in the evenings and weekends for training of Mentors. Classroom Teachers, the MathStrongLab Coordinator, and any additional 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 specifics of implementation of a MathStrongLab including methods of monitoring and evaluating students' progress.

 

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.

 

5.)              Scheduling of the use of the MathStrongLab

 

Classes should use the Lab for regular daily mathematics instruction.  Depending on the size of the school and the number of classes, there may need to be a MathStrongLab on each floor or in each wing of a school. 

 

In many schools, the MathStrongLab can be used after school and in the evenings and on weekends as well as during the school day.  After school and evening and weekend sessions can be used by the following:

 

§  Further progress for daytime students.  Extended day programs can allow students to go to the MathStrongLab staffed after school for additional time with the math manipulatives as well as with math-related games which provide further mathematics and logical stimulation and practice and also strategy development

 

§  Professional development for teachers and volunteers as MathStrongLab Mentors and Mentor Assistants,

 

§  Adult basic education

 

§  Professional development for personnel from other schools and home schooling parents

 

§  Experiences in the MathStrongLab for parents or caregivers with their preschool children.

 

6.)              Methods of Monitoring Students' Activities and Progress in the Lab

 

Of course, students' worksheet and problem solutions need to be checked.  And they need to be checked in real time rather than collected and checked later.  Checking of a student’s worksheet involves an interaction between the student and the Mentor.

 

Each student needs an opportunity for recitation.  After a student completes a worksheet, s/he raises his/her hand and a Mentor comes and has the student do some appropriate recitation concerning the completed worksheet.  This would include reading the equations or inequalities or fraction names or whatever other information the student generated on that worksheet.  Although we want students to discover whatever they can about mathematics, there is also a need to provide "cultural" aspects of mathematics, that is, terminology and notation.  For example, students need to be told "The numbers that we add are called 'addends' " and "The result when we add is called the 'sum.' "   An appropriate time for the Mentor to provide this information is after the worksheet is completed and the preliminary recitation is done.  Then this new terminology can also be practiced on that and subsequent worksheets.

 

Students file their own worksheets, with monitoring by a Mentor (especially for young children), in file folders labeled with the student's name and the set (e.g., "+ - C Rods" for Addition and Subtraction with Cuisenaire® Rods or “DF" for Decimal Factory).   This provides safekeeping of each student's work as well as a record of what work the student did on a given day since students are required to put name and date on each worksheet.   The Classroom Teacher can be responsible for recording each student's progress in each of the sets of materials (by looking at the front page in each file folder as well as periodically checking through the file folder to be sure worksheets are in order and checked) and communicating this to parents along with the topics covered in each set and how far the student has progressed through those topics.

 

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.  For example, the student may be exploring fraction concepts using Pattern Blocks, developing natural number concepts with Pegboards, and drill arithmetic at the same time as solving algebraic equations with Hands-On Equations®. 

 

 

Many will maintain that implementing a MathStrongLab is impossible due to the challenges, particularly of having the appropriate Mentor-to-Student ratio with properly trained Mentors.  However, education decision makers and citizens in general should reflect on the result of students coming out of Kindergarten through Grade 12 education relative to their competence in mathematics.  We need to realize the suffering of those who are pulled along with the class while not “getting” it and thereby come to feel frustrated by mathematics and remain unable to deal with everyday situations which call for the knowledgeable use of mathematics.  We likewise need to realize the time being wasted in school by those who are particularly talented in mathematics and are limited in the breadth and depth of their study of mathematics by staying with the class.  It should not be our intent to “narrow the achievement gap” in mathematics.  Our intent rather should be to raise the achievement of all and at the same time allow all students to appreciate the beauty and utility of mathematics as well as their own ability to understand and use mathematics.

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Fran Armstrong,
Dec 30, 2008, 1:48 PM
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