"I hate math! I couldn't learn it, and I can't teach it!".

"I'm bad at math; it's always been my least-liked subject!"

"I hated math in school . . . and my feelings haven't changed since."

"When my parents moved to a new town, the kids had been doing long division for a semester already and I hadn't done any. The teacher and everybody thought I was stupid!"

"I don't see why I have to teach math. I never could do it when I was in school!"

"I enjoyed math, and my dad and I played games to see who could get an answer first. I know now he let me win lots of times."

Those comments were made by graduate students taking a math instruction seminar for certification as elementary school teachers. They had been asked to respond to these questions:

* Did you enjoy math in elementary school?

* Did you do well in math in elementary school?

* Do you enjoy math now?

* Do you do well in math now?

The students were nearly evenly divided between those who liked and those who disliked math. In nearly all the cases, a correlation existed between attitude and success. Nobody claimed to have liked and done well in math as a younger student if they also presently disliked and did poorly in math. On the other hand, a small but significant number stated they had disliked and done poorly in math, but at some point, their ability and attitude had improved. This group, albeit small, provided an opportunity to explore what might account for the change. Participants were asked to identify factors they felt impaired their math learning, or that contributed to having a negative attitude.

WHAT ARE SOME SOURCES OF FRUSTRATION AND FAILURE?

Teachers' assumptions of students' knowledge. Many of my graduate students believed that their math teachers acted as if computational procedures and processes were simple and self-explanatory; even worse, students said, their teachers had little sympathy for students said, their teachers had little sympathy for students who did not understand the concepts. Indeed, it may be difficult for one whose interests naturally gravitate to things mathematical to recognize that seemingly simple, self-explanatory processes may be complicated to others. Gardner's (1983) theory of multiple intelligences posits seven areas of innate propensity: linguistic, logical-mathematical, spatial, musical, bodily-kinesthetic, interpersonal, and intra-personal. These intelligences are not mutually exclusive, and individuals may display strength in some and weakness in others. For example, a 5th-grader may seem unable to compose a simple paragraph, yet is eagerly sought whenever someone's VCR has defied all efforts to make it function, or an erudite poet or philosopher cannot balance a checkbook. Most of us recognize variations in ourselves. An anecdote comes to mind in which the renowned musician and composer Leopold Stokowsky is reported to have been coaching his equally renowned friend, physicist Albert Einstein, as the latter attempted a simple tune on the piano: "No! No! It's one-two, one-two-three . . . and . . ., Good Lord, Albert, can't you count?"

Those with "logical-mathematical intelligence" (Gardner, 1983) have a talent for manipulating numbers. While this may constitute a small group, the poet can learn to balance a checkbook. It may, however, take more patience and a slower pace than a natural mathematician would find comfortable.

Obscure vocabulary. Too often, unique math vocabulary is not explained, such as "divisors," "integers," "quotients," "multipliers," "differences," "products," "multiplicands," "minuends," "subtrahends," and other math-specific terms. Even adults who use math daily may have difficulty defining all of these terms. Yet, they abound in textbooks and are used frequently in instruction. Teachers must take time and care to explain these terms, and to ensure that students understand them, before launching into explanations of their interactions. A child struggling to distinguish between a quotient and a divisor is not ready to understand an explanation of long division.

Incomplete instruction. Students often were frustrated by a lack of instruction in the sub-steps of mathematical procedures. To explore individual computational strategies, or algorithms, seminar participants were asked to "add a column of figures" [ILLUSTRATION FOR FIGURE 1 OMITTED] and explain in detail how he or she had arrived at the answer. Their computational algorithms showed marked distinctions:

a. Begin with the column on the right; look for groups of 10; add the "10s," then add anything left over. Put down the "ones place" number and carry the "10s place" number to the second column; repeat the process for each remaining column.

b. Add the top pair of three-digit numbers, and put the sum to their right. Repeat the process for each pair of three-digit numbers as you progress down the column. Finally, add the "sub-sums" at the side.

c. Begin with the column on the right, add the first two numbers, and write their sum to the right. Take the next two numbers in the column and repeat the process, and so on down the column. Add the "sub-sums" on the right. Put down the "ones place" number and carry the "10s place" number to the next column; repeat the process for each remaining column.

d. Beginning with the column on the right, add the numbers in sequence from top to bottom, keeping a running tally in your head (4+1 = 5, 5+9 = 14, 14+6 = 20, etc.). Put down the "ones place" number and carry the "10s place" number to the second column; repeat the process for each remaining column.

e. Each digit is worth a specific number of"beats," as in music. Your ear detects if you have afforded a number the correct amount of "beats." Begin counting "beats" at the top of the right-hand column and move down the column. The total number of "beats" is the sum for that column. Put down the "ones place" and carry the "10s place" numbers. Repeat for each remaining column.

f. Add all digits in the left column and enter the sum in the first row below the line, plus two zeros for place keeping. Next, add the digits in the middle column and enter the sum in the second row below the line, adding one zero for place keeping. Finally, add the column on the right and enter the sum in the third row below the line. Adding the three "sub-sums" will yield the final sum [ILLUSTRATION FOR FIGURE 2 OMITTED].

Most of the students used method "a," and labeled the remaining strategies as "archaic," "confusing," or "cumbersome." Yet, well over half used one of the other methods, which they claimed to have learned or developed in elementary school and continue using to this day. Most effective strategies had been taught proactively in class, while the ineffective ones had been developed in the absence of effective instruction. Although some students admit that strategy "f," which was suggested by a student from Argentina, seemed effective, they found it "too difficult to explain."

Phillipp (1996) exhorts teachers in the United States to be tolerant of alternative, and equally effective, computational strategies, including those used by students from diverse cultural/national backgrounds. The National Council of Teachers of Mathematics (NCTM) (1989) suggests that teachers de-emphasize memorization of specific rules and algorithms, and instead encourage students to create their own algorithms and procedures. Still, teachers need to guide their students; otherwise, they may end up creating strategies that work temporarily, but will be ultimately ineffective. The teacher should provide suggestions before the child becomes frustrated, yet show tolerance toward effective variations. Such an attitude requires that teachers be able to identify an inefficient method and tactfully model a better one. Therefore, teacher training in math instruction must go well beyond "accepted" algorithms and procedures.

Too many skill-and-drill exercises. The graduate students recalled being given too many homework problems, and too many of the same kind of problems. The course participants believed that six problems should suffice for understanding, while endless repetition only led to frustration and increased math anxiety. Constructive homework, they add, should include a variety of applications, including word problems.

Frustration at not being able to keep up with the class. My students remembered the frustration of trying to keep pace: "I tried to follow and understand, but I couldn't keep up with the teacher," "I'd almost get an idea, then the class would go on to something else," and "I'd get one or two concepts or processes, but lots of times four or five things would come at once." Math courses typically depend upon learning and understanding certain preliminary material upon which subsequent learning is built. Thus, keeping up with the class is critical. Success in math is analogous to a foot race. Once you fall behind, catching up - let alone winning - becomes progressively less likely. Thus, a math teacher needs to recognize when students become "lost," and then initiate timely corrective action. Perhaps one reason why remedial math classes generally do not boast high success rates is the long period of waiting between the time one falls behind and the time spent on re-learning concepts. It is much more effective to address a student's problems as soon as possible, which also helps reduce frustration and anti-math feelings. One of the major premises of the so-called Saxon method (Larson, 1991) is incorporating continual review into the curriculum by having students return to concepts covered in previous lessons. This approach appears to help students identify concepts, strategies, or techniques that they found difficult.

An overemphasis on rote memory. Ironically, few math teachers stress memory as a major factor in learning math, nor do they believe that memorizing formulae or processes is the best approach (National Council of Teachers of Mathematics, 1989). When concepts are understood, memory becomes but one of several tools employed in problem solving. Interestingly, dependence upon memory is more often a student-initiated phenomenon, and occurs typically when basic concepts are not understood. A colleague who teaches university mathematics notes that many of his less successful students are reluctant to delve into questions of why a particular strategy, formula, or algorithm works, or to consider alternative strategies. Instead, they prefer to be given a formula they can memorize and apply to all problems of a certain type. Such students, he notes, not only rob themselves of a basic understanding of math concepts, but also frequently reach a point of "memorization overload."

Math instruction presented in isolation, as an end in itself, and with little tangible relationships to the real world. The rationale behind such instruction seems to be that students will learn mathematical operations, then "store" them for future use on a test, much like is often done with vocabulary lessons. While teachers may assume that students will be able to identify and recall the appropriate operation when needed, in reality, many students will not be able to do so. Among undesirable byproducts is a high rate of forgetting over time. Beyond simple calculations, one often cannot connect a specific mathematical operation to a need. At best, the situation may trigger a vague recollection of a procedure that seems related but whose specifics are evasive. The fortunate person may convert this recollection into a systematic search that results, in essence, in re-learning the operation in order to apply it.

What Is Effective Math Instruction?

Effective instruction is reflected in six key questions, the answers to which provide insight into an instructor's focus and teaching style.

What is the purpose, or purposes, of math instruction? Does it seek to develop a student's ability to search and gain access to mathematical techniques, and then appropriately apply them to real-life situations? Is the goal to expand students' mental capacities through challenging explorations into mathematical theory? Is it important that the student develop an interest in and appreciation for mathematical processes? Or, should the focus be upon teaching students to perform various mathematical calculations quickly and accurately, while identifying and separating those who display a capacity for logical-mathematical thinking (Gardner, 1983) from those who do not?

Although performing calculations quickly and accurately is useful, effective instruction emphasizes skill development in the context of meaningful applications to real-world situations, while cultivating students' interest and confidence in using math.

For whom should math instruction be designed? Do "serious" math students bring certain skills, capacities, or talents to the classroom? Are there students for whom math instruction is probably a waste of time? Is our goal to increase the math capability of every student in class, or just for those who show talent and interest? Effective instruction should focus upon all students, and aim to increase the math capabilities of each, regardless of what talents they may - or may not - appear to possess. Encouraging math instruction may rejuvenate a child whose ability initially seems limited, or who may have been "turned off" by poor instruction or negative experiences.

What determines the scope and focus of a math lesson? Traditional math instruction typically transmits concepts and processes according to a timetable pre-set by the teacher, the textbook, or local administration, and lends support to the concept that students must keep up or fall by the wayside. Teachers often begin lessons with a textbook explanation of an algorithm or process, continue with independent practice through worksheets and homework, and culminate with a test on problems using the process. The textbook is often the primary determinant of learning material; therefore, acquiring those skills often is incumbent upon reading the text and doing end-of-chapter problems.

Such a textbook-driven curriculum may contribute to students' falling behind, failing to see how math relates to their lives, and depending upon rote memory rather than reasoning. By contrast, beginning a lesson with a real-life situation, continuing with guided exploration of various mathematical techniques, and culminating with students applying those techniques to achieve solutions will capture students' interest and attention. Here, the text becomes a reference, rather than the determiner, of the lesson or student success.

What instructional activities and materials should be incorporated into a math lesson? Are textbook exercises and workbooks or worksheets all that one needs to teach math? How much classroom time, if any, should be spent in hands-on practice in problem solving? Should students solve problems at the chalkboard? Should they work in cooperative groups? For younger children, should manipulatives such as blocks, beads, variable-length rods, and other devices be used? What place should measuring tapes, thermometers, scales, and other instruments have in the classroom? Do calculators have a place there?

Textbook exercises, workbooks, and worksheets are rarely stimulating, they typically focus upon calculations in isolation and in the absence of meaningful context, and they can contribute to students' perceptions of math as being irrelevant to their lives. By contrast, hands-on exercises and real-world measuring devices add meaning and relevance in an interesting context. Grossnickle, Perry, & Reckzeh (1990) note the importance of developing a student's ability to relate math to real-life requirements, which, in turn, gives meaning to mathematical operations, as well as enhancing learning and a positive attitude. Likewise, NCTM's standards for mathematics instruction (1989) emphasize focusing more attention on meaningful relationships and

applying problem-solving strategies to realistic situations, while decreasing the amount of instruction devoted to learning paper-and-pencil algorithms without a context. Working in groups tends to enhance individual motivation and provide opportunities for alternative learning or teaching styles, such as peer tutoring. Using calculators in the classroom does not devalue instructional objectives if the objectives focus upon understanding mathematical concepts, rather than simply performing specific algorithms.

What form, or forms, of assessment should be employed, and how should they be used? My seminar participants recounted how most of their math lessons culminated in the inevitable test, which featured two unsavory elements: it weeded out those who could solve the problems in the time allotted from those who could not and, more worrisome, the test became virtually an end in itself and the prime motivator for study. Most participants recalled that such tests strongly contributed to math anxiety, undermined self-esteem, and reinforced their instructor's - and often their own - low opinion of their math capability. NCTM (1989)has encouraged moving away from a test-dependent assessment program that focuses upon students reproducing discrete calculations in isolation. Instead, it advocates programs that give students multiple opportunities to demonstrate their mathematical powers in meaningful contexts.

How should errors be identified and corrected? Should students be encouraged to ask questions? If so, how should they be handled? What, if anything, should be done if students fail to solve problems correctly? Seminar participants asserted that the manner in which an instructor identifies math errors closely correlates with the student's self-esteem and attitude towards math. Teachers should identify students' errors, and treat their questions in a non-threatening and non-personal manner. Students' questions, after all, can provide invaluable clues to their understanding of the material.

Many seminar participants reported that their elementary school teachers rarely tried to determine the reasons for students' difficulties with certain operations or processes. As these teachers saw it, problems either were solved correctly, or they were not. Fortunately, an increasing number of today's teachers are analyzing not only test results, but also homework and other forms of problem solving, with an eye to identifying recurring errors. When such patterns are found, the instructor can focus on specific remediation. By contrast, this author recalls a well-meaning teacher once asking him, "What, exactly, don't you understand?" Unfortunately, someone who is confused and overwhelmed rarely can identify the specific locus of misunderstanding. Now, however, diagnostic error analysis often can pinpoint and remedy a misunderstanding (Ashlock, 1994).

An Example of Mathematics Instruction in a Meaningful Context

Students in a 6th-grade class in Honduras were having trouble learning to calculate angles, dimensions, proportions, areas, and volumes (Cornell, 1989). Their test scores were low, and their math anxiety ran high. Using textbook explanations and pencil-and-paper exercises proved fruitless. The students' frustration and hostility grew evident, affecting classroom control.

In desperation, the teacher reoriented the curriculum by first changing the textbook's role from curriculum driver to useful reference, and next, proposing a real-life project - constructing a playhouse for the kindergarten. Students were given responsibility for all aspects of planning, design, and construction.

The students attacked the project with vigor. They brought carpenters' squares and measuring tapes from home. Angles, squares, and parallels became critical to laying out plot and floor plans, which, of course, had to be drawn accurately and to scale. The Pythagorean theorem took on new significance. Students in charge of determining material requirements needed to know how many bricks would be needed...and how to allow for the mortar spaces between them. They also needed to know how many floor tiles, 8 1/2 inches square, go into a room 6 by 8 feet.

The lessons began with identifying needs. "For a stairway, how many steps can we put in the space we have available and how high should each step be?" The project gave meaning and purpose to math, and the students assumed responsibility for classroom control. The teacher still gave explanations, but usually in response to students' questions, such as "How do I account for the windows when I'm calculating how many bricks I need for this wall?" Each of the work crews discussed their progress, exchanged ideas or suggestions, and engaged in "trade-off" studies with the whole class, evaluating the pros and cons of a particular design. The assigned homework was far more interesting than traditional assignments. The hands-on use of measuring instruments lent purpose to object manipulation.

These students were not exceptional nor did they possess special talents in math. Although logical-mathematical skills were emphasized, verbal and written descriptions, group interpersonal interactions, drawing abilities, spatial perceptions, and many other skills also were needed. The assignment called upon an expanded range of skills and a variety of "intelligences" (Walters, 1992). At term's end, the students' understanding of mathematical concepts, as well as their ability to perform calculations, far exceeded 6th-grade norms.

Why was this project so successful? The nature of the project itself was not critical. Building a piece of furniture or a miniature boat, or constructing an Indian village or a doll house to scale, would have provided the same stimuli. Rather, success depended upon the teacher's willingness to reassess the purposes of instruction and to modify the curriculum to increase students' involvement without jeopardizing instructional goals. This modification required a re-evaluation of what students should learn from a math class - as well as sober reflection on what they should not have to learn (Buxton, 1991; Martinez, 1996).

What Factors Can Change an Individual's Interest and Success in Math?

Let us return to the small group of participants from the beginning of the article who had experienced difficulty with math in elementary and middle school, yet eventually underwent a change in both attitude and ability. One young man, when learning plane geometry, with its multitude of theorems, axioms, and corollaries that had to be "proven" to the teacher's satisfaction, did not know why he should care that opposite angles made by intersecting lines are equal or that parallel lines produce equal angles when intersected. Years later, however, while doing carpentry with his father-in-law, the abstract calculations of angles, radii, and line projections suddenly became recognizable among the templates, peaks, pitches, and eaves. Back he went to his geometry book, and other forms of math, which he discovered were not only useful, but also fascinating.

Another student, during 5th grade, spent two months at a different school. Having done three-digit multiplication at his old school, he shot to the top of the class at the new one, which was just beginning these problems. Things went well . . . until he returned to his former school. During his absence, his classmates had been unlocking the mysteries of long division. Suddenly, math became associated with frustration and failure, and it became a subject to be avoided. Later, in college, he needed to take a statistics course, which had college algebra as a prerequisite. After three attempts at algebra, he finally passed and enrolled in statistics. Initially, the complex formulae and multiple calculations discouraged him. Then, he found he enjoyed using statistics to express relationships and correlations in his world. He ended the term with a B+ and a new attitude towards math.

Another participant entered teacher training after raising her children. Math had been her nemesis in school. She was determined, however, that her students would not be as unsuccessful and resistant to math as she had been. Coaching them required a certain amount of re-learning on her part, plus a pretense of enthusiasm. As she developed her ability to convey math concepts, her enthusiasm became real.

What accounted for such changes in attitude and proficiency? In each case, math acquired a meaningful purpose. Absent was the threat of ridicule by teacher or peers. Gone was the anxious race against the clock associated with test-taking. Perhaps most important, each experienced success with math, which contradicted the "I-know-I-can't-do-it-so-there's-no-point-in-trying" syndrome a syndrome as frustrating to the conscientious teacher as it is to the unsuccessful student.

Teaching Math Effectively

What conclusions can be shared for improving both math instruction and teacher training?

* Instruction must include discussion of the real-world applications of the operations, calculations, and processes being taught.

* Personal ridicule, direct or implied, has no place in math instruction.

* Incorporating projects (large or small) provides meaning and purpose, adds interest, and helps develop intrinsic motivation and discipline.

* Math instruction should be interesting and fun, not "fun and games" lacking substance; nor should the content be dumbed down. Rather, it should reflect the idea that learning and motivation are enhanced when students can enjoy stimulating math-related projects, contests, demonstrations, and similar activities.

* Math vocabulary should be explained, not assumed. This will save time in the long run.

* Rote memorization exercises should be deemphasized; furthermore, a student's dependence upon rote memory should be viewed as a possible sign of gaps in understanding.

* Instruction should incorporate an element of diagnosis and remediation that detects and addresses student errors as they occur. This aids students, while informing the instructor of unlearned concepts or operations that need re-teaching. Formative as well as summative evaluation occurs as students address and correct errors while in the process of learning. Thus, remedial math becomes a part of the regular curriculum, rather than something only those who have failed are entitled to receive.

The twin tasks of increasing the effectiveness of math instruction and ensuring a pool of capable and enthusiastic teachers means considering both content and affective factors, as well as a willingness to take risks where necessary to give meaning, purpose, and relevance to math instruction. As attested to by the seminar participants, this is particularly critical at the elementary level, when gaps in understanding and feelings of frustration and negativity often begin.

References

Ashlock, R. (1994). Error patterns in computation. New York: Macmillan.

Buxton, L. (1991). Math panic. Portsmouth, NH: Heinemann.

Cornell, C. (1989). El Alba papers. Unpublished records of El Alba School. Siguatepeque, Honduras.

Gardner, H. (1983). Frames of mind. New York: Basic Books.

Grossnickle, F., Perry, L., & Reckzeh, J. (1990). Discovering meanings in elementary school mathematics (8th ed.). Fort Worth, TX: Holt, Rinehart and Winston.

Larson, N. (1991). Math: An incremental development. (A K-5 series.) Norman, OK: Saxon Publishers.

Martinez, J. (1996). Math without fear: A guide for preventing math anxiety in children. Needham Heights, MA: Allyn and Bacon.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

Phillipp, P. R. (1996). Multicultural mathematics and alternative algorithms. Teaching Children Mathematics, 3, 128-33.

Walters, J. (1992). Application of multiple intelligences research in alternative assessment. In Proceedings of the Second National Research Symposium on Limited English Proficient Student Issues: Focus on evaluation and measurement, Vol. 1. United States Department of Education, Office of Bilingual Education and Minority Language Affairs. Washington, DC: United States Government Printing Office.