Teaching Mathematics Effectively

Robert D. Hackworth, Educational Consultant

Unless we produce students who are active, thoughtful learners, we are simply pushing them up a greased pole.

Curtis Miles

One of the procedures I was taught long ago is that a teacher needs to preview a lesson. I believed that. Also, I was told that a presenter needs to set an agenda for the audience. I believed that too. And when they came along, I was thoroughly convinced of the value of behavioral objectives: a teacher is obligated to state specifically his/her purpose of instruction and communicate the objectives to the students. Again, I believed what was claimed about behavioral objectives.

Today, I am less enthralled with behavioral objectives. I agree completely with the idea that each instructor should have prepared well for each lesson, but I am today less certain about the specificity that is needed. I was taught that the only acceptable behavioral objectives were those that could be measured, and today I feel we have trivialized many objectives just so they can be measured.

At this point, you should be wondering where I am going. That's part of my strategy for this presentation. First, I want you to know that I'm pulling old belief systems out of the education closet and seeing which ones should be discarded. Second, I want you to know that I have planned carefully for this presentation with the full confidence that if you pull me away from my original objectives, then I can effectively deal with that situation.

Before I leave behavioral objectives completely, there is a second problem about them that also violates the latest thinking in mathematics education. To some degree, the practice of informing a student at the beginning of a lesson of its purpose is being questioned. The idea that a student should be encouraged to accept a preordered organization of concepts and skills contradicts some of the current beliefs about the most effective way to learn mathematics.

This is all prelude, of course, to my hope that you will not feel restrained by your own belief that I might be doing something so important here that you should not interrupt with questions or comments. I am, I think, well prepared with my own script, but I shall be delighted if we find some other paths to explore along the way. As you know, a paper copy of what I have planned will be included in the proceedings of the Institute, so the information I have planned to bring you will be available anyway. That includes all of my overheads, so there is really no need to take notes. I encourage you to sit back, listen, think, and ask questions.

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The first order of business we need to address here is the tension, maybe outright hostility, that exists between Mathematics Departments and all other departments within our institutions. In truth, there has been very little communication between these warring camps for years. Most attempts to communicate are frustrating exchanges in which different value systems make the best of ideas fly by the other side without ever making contact. Nothing we do here today is going to solve this difficult problem, but I assure you that progress in mathematics education cannot be achieved without some improvement in our communication. There should be no adversarial relationship between mathematicians and non-mathematicians. I am not your academic enemy, even though I may have credentials that may indicate it. You are certainly not an enemy of mathematics, even though some of the problems I need to talk about may touch programs close to your heart.

Consider me a bridge between the mathematics establishment and yourselves. My experiences have put one foot in each camp, and when I speak of the hostility between them, I remember the heat I have taken from both sides.

I began my teaching career in mathematics with a plan for my work life that is not recognizable when compared to my actual experience. I still counsel my offspring and my students to plan ahead, but it is only with the belief that some planning will better prepare one for facing the drastic changes that are, in my experience, unavoidable. Please treat the advice I give you today accordingly. I would probably give different advice if asked tomorrow and undoubtedly would do so next year.

I plunged into developmental education by accident. My first steps toward a lifetime's work were taken without any intention of commitment. Florida in 1966 was making a serious (that means well-funded) attempt to deal with students unprepared for any college mathematics course, and I volunteered to participate for one-third of my class load. My responsibility was to teach a class of eight students! The mathematical content was beginning and intermediate algebra. Obviously, we believed that small classes with a great deal of personal attention for each student would be effective. It was. It would be today. And, of course, it is no longer considered economically feasible EXCEPT, as some of you realize, with some student tutoring programs.

Returning to my own plunge into developmental mathematics, when the first economic crunches began, our successes with small classes and personal attention seemed too valuable to simply dismiss. When compared to the frequent failures associated with traditional programs of lecture or lecture/discussion, our positive experiences made us doubt the efficacy of even those instructional programs, like calculus, where the failure rate was "acceptable." But we began with developmental mathematics because that was the point in the curriculum where mathematics professors allowed some, not much, tinkering with instruction.

And this is a good point to consider the meaning of "instruction." In the late 1960's, I had a fairly simple concept for instruction which limited it to "delivery systems." How could I best "deliver" knowledge to my students? Issues of motivation, attitudes, study skills, goal setting, were students strengths/weaknesses which I knew had a serious impact on learning,

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but at that time they were separate from "instruction." Consequently, my first attempts to improve mathematics instruction focused on delivering knowledge to students in ways that would make the material easier to grasp and master. Examples of improved delivery systems included the development of better explanations in more logical sequences, more information available in print materials, mastery testing materials, and expanded use of video and computer materials. Results of those efforts were gratifying, and by the end of the 1970's, I was approaching Bloom's criteria for excellence: 90% of my students were achieving 90% of our course objectives.

Then the numbers began to drop. Delivery systems, by themselves, were no longer sufficient to guarantee quality learning. The reasons for the drastic fall-off in mathematics learning are complex (social, political, educational), but the effects are familiar to all of us trying to teach, tutor, and/or manage instructional programs. Students are failing mathematics in large numbers. Fear and anxiety of mathematics is commonplace. Complaints of mathematics instruction abound. Serious questions of the relevance of mathematics are raised even in the face of technology growth that demands greater not less understanding.

Meanwhile, the vast majority of mathematics professors are today defining our instructional problems in the same terms as they were described in the 1960's. Most mathematics teachers continue to look for the WONDER BOOK or WONDER COMPUTER PROGRAM that will miraculously change student achievement levels. However, the leadership of the major mathematics associations in the country have shown a far deeper understanding of the problems and are accepting major responsibility for addressing it. The National Council of Teachers of Mathematics (NCTM) has courageously attacked the teaching practices of its members. In 1991, NCTM published its Professional Standards for Teaching Mathematics that promotes a different vision of mathematics teaching. The Standards states in positive terms the nature of the subject and methods for teaching it appropriately and effectively. No reader of the Standards can avoid its underlying criticism of current mathematics instruction. There are today more articles in the mathematics journals concerning instruction or learning than I have ever observed in the past. And change is coming. It's slow too slow but improvements in teaching mathematics have begun.

This last week one of the employees at my school said they really liked working when there weren't any students around. That's because students are a problem, and student attitudes toward mathematics are a serious obstacle to any improvement in the services we provide. Large numbers of students who seek help with mathematics have a very different idea of "help" than those of us who are to provide it. Conversations with such students rarely are instances of good communication. Excellent explanations are frequently ignored as useless ritual. The student waits for the rote procedure needed to get answers. Mathematics teachers see the explanations as the "true mathematics" and the procedures used to actually solve problems as following logically. Meanwhile, the students see the procedures as "true mathe-mat-ics" and the explanations as unnecessary fluff. These students try to pass the course by memorization. Often, they make Herculean efforts and still fail because those efforts are so inappropriate for learning mathematics.

It is easy to place the blame on the students, but it is also obvious that they learned to treat mathematics this way. Now, as professionals faced with this challenge, what do we do

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about it? Some mathematics professors and tutors seem oblivious to other sources for the learning problems and explicitly state that the problem is one for the student to overcome. Other professionals recognize that the difficulties are more complex, but have no strategy for teaching students more effective ways of learning mathematics.

When minds are stuffed with knowledge they don't understand, their thinking becomes chaotic.

Kamii & DeVries, 1978

Major blame for student failures in mathematics can be properly placed on their education. Teachers, school systems, social myths, and public policies have created these problems, and they will be with us until they are eliminated from elementary/secondary school practices. This indictment of elementary/secondary school practices does not mean that we in higher education are any better (maybe worse), but the problems begin earlier than we see them, and solutions must be implemented earlier in our education systems. For us, we can expect at least ten to twenty years more of accepting new students who have learning deficiencies caused by their past schooling. The extent to which we deal wisely and kindly with these individuals will have a major impact on their intellectual achievements across the curriculum.

Most education "reforms" have been failures. Longer school days, more days, more required courses, standardized tests have not brought improvements. My newspaper last Friday announced a study that American students continued to compare poorly against most others. Although I think the standardized testing craze is part of our problem, I am pleased for the results that indicate our reforms have been a failure.

Some current efforts are not only ineffective; they are self-defeating. In some instances, mathematics professors and tutors, under criticism from students and administrators, have attempted to find better ways for their students to learn mathematics by rote. Materials and testing procedures have been altered accordingly. Students in these situations may overcome the short-term hurdles they face, but the long-term results are disastrous. In other words, they pass their mathematics requirements, but are illiterate in the subject. Sadly, it is many of these students who have become elementary teachers, or other decision-makers in education, and have promoted the wider use of practices which greatly enlarge the problem.

As a first step toward improving our teaching of mathematics, I would require all of us with responsibilities in the subject to spend some time and effort acquiring an insight into the nature of the subject. There is a startling disparity of thought between mathematicians and those who have trouble with mathematics. That disparity effectively blocks most communication. When I say, "Mathematics is fun," many of my students reply, "That's because you know all the rules and we don't." When a student says piously, "Mathematics is very important," I am reminded of Henry Whitehead's statement that some avocations deserve a special position because of their intrinsic worth, and then he cited music, mathematics, and the making of good shoes!

Peter Hilton, distinguished professor of mathematics at SUNY Binghampton describes the nature of his subject in this way:

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Mathematics is systematized thought, supported by a beautifully adapted language and notation. It is characterized by the recognition, discovery and creation of pattern, and by the establishing of subtle connections between its apparently very dissimilar parts. Contrary to traditional school practice, it is not a set of distinct subdisciplines, but a unity, drawing on a diverse but interrelated repertoire of concepts and techniques. Again contrary to popular belief, it is not a set of facts; and mathematical understanding is not to be measured by tests of knowledge and memory. Thus, for the student, what matters is that he or she learn to think mathematically and any significant part of mathematics can be used as the vehicle to convey the necessary understanding and thinking ability. Conversely, no part of mathematics, however seemingly appropriate, can prepare the student really to use mathematics intelligently and effectively, if it is taught simply as a set of isolated skills, to be retained by the exercise of undiscriminating memory.

Joe Garofalo, professor of mathematics education, University of Virginia describes mathematics in terms of the student outcomes we should expect:

I want students to develop mathematical power, meaningful concepts, healthy beliefs about the nature and value of mathematics, confidence in their ability to learn and use mathematics, and useful problem solving strategies. Traditional teaching methods are not very effective for helping students achieve these goals; actually such methods often work against them.

It is my experience that most students having trouble with mathematics focus on retaining subject matter content without first learning the intellectual skills needed to support that effort. Consequently, a second major thrust toward improving mathematics education must involve a broader view of instruction. I stated earlier that instruction has been viewed as a "delivery system" which ignores many factors which impact on learning. Today's quality instruction must take responsibility for these other factors. Claire Ellen Weinstein, professor of educational psychology at the University of Texas, states that learning has at least three facets that must be addressed by instruction:

1. Skill. The language, techniques, facts, etc. of a subject.
2. Will. The attitudes and motivations which drive efforts to learn.
3. Management. The ability to make appropriate decisions for engaging in learning activities.

Obviously, most traditional instruction devotes little time or effort to two of these facets. In fact, all planning is focused on content delivery. Course calendars are written in

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terms of content coverage, and serious criticism is given the teacher who does not "cover the required curriculum."

When the instruction does attempt to address other issues besides content, there is often a superficial quality about those efforts. We have had few experiences, in school, upon which to model such instruction. In fact, the best models for such instruction are found outside our schools.

Some years ago, Claire Ellen Weinstein outlined four areas of concern for quality instruction. She claimed that they were necessary and sufficient conditions for learning. I have applied them to the teaching of mathematics and have found them to greatly improve my awareness and monitoring of my own instruction. When I am aware, I include all four areas in my instruction. When I evaluate my instruction, I find that using the four areas gives me an excellent model for monitoring it. I believe that instruction of mathematics based on these four areas of concern can meet the views of mathematicians about the subject and also conform with theories of learning. We need to do both.

1. Create Quality Learning Environments

Each of our students lives in two environments: an academic environment and a nonacademic environment. Most students in community colleges spend the greater part of their waking hours in non-academic settings, and many of the difficulties for those students are directly related to influences outside the campus boundaries. Those of us who work in institutions where most or all of the students live on campus are blessed by circumstances where the on-campus problems generally dominate. I'm going to restrict my comments to the academic environment, but that is due, in part, to my helplessness to suggest ways of overcoming problems in the non-academic environments.

Our premise is that what a student learns depends to a great degree on how he or she learned it.... for each individual, mathematical power involves the development of personal self-confidence.

Curriculum and Evaluation Standards for School Mathematics

For some of you, your workspace is on a campus with real academic character. Others here, no doubt, work on campuses which were built on a warehouse model. In my teaching situation, it seemed that the same architect designed all the colleges and used the same design he had used earlier for the prisons.

Students tend to transfer their memories of past mathematics learning environments to new situations. Frequently, we make that easier because most math classrooms do look alike: a blackboard with rows of desks. Make the environment different from what the student has experienced in the past. Encourage the student to approach this learning experience differently by changing the physical qualities around them. Quantitative changes like less blackboards, less desks, are helpful, but qualitative changes are vital. Look for ways to raise the cultural level.

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1. Hang plants in the room.
2. Play background music (preferably classical with slow beat).
3. Hang paintings from the college library.
4. Put flowers on the tables.
5. Carpet seating areas.

The major environmental factor for the student will be the teacher, tutor, and/or manager. Break the stereotypical image of a math person. Students expect mathematicians to be cold and logical. Surprise them; be warm, friendly, and maybe illogical. Students often believe mathematicians have no other interests; share the fact that you have a life outside mathematics:

1. Bring a novel you are reading with you to class.
2. Talk about a concert or play you recently attended.
3. Mention characteristics of parents, friends, that made you mathematical.
4. Be personal.

Play down the role of education for making money. That idea is far oversold, and the student already gets plenty of it. Instead, emphasize the historical, cultural, intellectual role of mathematics. All content makes good sense under that umbrella. Puzzles, ridiculous word problems, rationalizing denominators in the age of calculators, are wonderful topics when we get past the argument of "How am I ever going to use this?"

Anxiety is a major problem for many mathematics students. Recognize their fears and publicly admit to being lucky that our own experiences were unlike theirs. Avoid scare tactics and harsh rules because they will not work with students who are already afraid. Exude patience, confidence, and a belief that hard work under your direction will overcome. Emphasize that anxiety frequently interferes with properly budgeting time, persevering, asking questions, and taking responsibility for their own learning.

Most students in trouble are reactive rather than proactive. They may attend class religiously, take prodigious quantities of notes, and strenuously attempt to decipher those notes. If so, they deserve an "A" for effort and will probably earn a low grade in mathematics.

To become proactive your students should be taught Benjamin Bloom's three factors for predicting success/failure. Bloom's first and major predictive factor is the amount the student knows before a topic is taught cognitive entry skills. Those who know the most at the beginning almost always know the most at the end. This is often a prediction of failure for weak students. In fact, it is a strong indication of what needs to be done to be successful: Prepare before each session, and learn as much as possible about the topic before instruction begins.

Bloom's second factor is affective entry skills. Affective domain factors such as motivation, beliefs, values, and attitudes invariably facilitate and/or debilitate thinking and

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learning. Occasionally these affective entry skills dominate all others. This is the case with students suffering severe anxiety or with those who are unable to accept a mathematician's view of the subject. In general, however, affective entry skills are of minor importance compared to prior knowledge. Another important aspect of affective domain factors is the fact that they often appear to be less under the control of the student. In fact, however, good students do improve their probabilities of success by their awareness of strengths or weaknesses in this area.

The last major factor listed by Bloom is teacher behavior, and the research can be most discouraging for teachers. It seems to indicate that we have a minor influence on success/failure. In many instances that is true, but the teacher who uses Bloom's research can reverse that negative outcome. The teacher that assesses cognitive and affective skills, and then actually does something about them, can use them to positively influence outcomes. For example, since cognitive entry skills are so immensely important, quality instruction must always begin where the student is rather than where the course assumes she is.

Students who are aware of Bloom's factors can also see themselves as responsible for success/failure an absolute necessity for any instruction to be effective. They can regulate their study to learn more effectively. They can also voice some needs that must be addressed if they are to succeed.

2. Process Information Correctly

Mathematics information can be divided into two categories:

1. Information that needs to be carefully memorized.
Examples are definitions, symbols, postulates, perhaps some formulas, and occasionally a rule that needs to be practiced before it can be understood.

2. Information that needs to be figured out each time it is encountered. Examples are procedures, rules, and problems.

Good mathematics students use these two categories correctly. Poor mathematics students practice them in exactly the opposite way.

The current rage in mathematics education is Constructivism. The allure of this psychological theory is that it emphasizes prior knowledge and making inferences. Mathematical knowledge under this interpretation is developed like a brick wall. The wall is made of bricks, but the integrity of the structure depends upon the positioning of the bricks and the quality of the mortar that connects them. The knowledge of mathematicians is organized like the bricks in a wall, but the knowledge of students having trouble with mathematics is more like the bricks in a pile, where each brick represents a separate entity, and there is neither organization or connections.

A word that is used frequently in discussing mathematics achievement is "understanding," but when most of us are asked to explain what that means, the results are often so

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vague that any teacher can claim quite honestly that they teach for "understanding." The constructivists have given us a better way for evaluating our teaching.

Mathematics information is normally hierarchical. Often, there is a logical learning sequence which will make learning easier.

Mathematics information is always interrelated. Regardless of the topic, it has relationships with all other mathematics.

With few exceptions, new topics in mathematics can be described by concrete situations. The famous psychologist Piaget strongly suggests that all new topics be introduced in that manner and later translated to more abstract situations. Some teaching and learning violates this Piagetian principle. For example, many elementary texts teach a method for finding the least common multiple of two numbers that is, for the student, a bit of magic. If that student first listed multiples of pairs of numbers and selected the least common multiple from the lists, the magic would be unclothed and seen as a shortcut for the more understandable process.

Besides using concrete situations to introduce abstract concepts, a second requirement would be to provide time and opportunity for the student to process new topics. The psychologist David Kolb describes the amount of learning as the addition of two vectors: one vector is the amount of instruction, and the other perpendicular vector is the amount of processing. If you double the processing,then you double the learning.

One of the most valuable understandings that needs to be explicitly taught is the existence of three types of knowledge in mathematics.

1. Declarative Knowledge The Whats of Learning
2. Procedural Knowledge The Hows of Learning
3. Conditional Knowledge The Whys and Whens of Learning

Good mathematics students treat the three types of knowledge as equals or, if not, they treat #1 and #3 as the most important. Poor students generally are unaware of the different types and, when they are, place great emphasis on #2. This means, of course, that they are making learning more difficult and less enjoyable. Teachers can help their students with these forms by constantly modeling by asking questions such as:

"What" questions that seek declarative knowledge:

1. What do I know about this?
2. What am I trying to find?
3. What can I do to present the problem another way?
4. What part of this problem can I solve?
5. What steps/strategies will I use?
6. What does this problem mean?

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7. What sort of answer might I anticipate?

"Why" questions that seek conditional knowledge:

1. Why is this answer reasonable?
2. Why does this work?
3. Can I explain my strategy to someone else?
4. Is there another way to solve this problem?
5. Why is this the best approach on this problem?
6. Why would someone make errors on this type of problem?
7. Why would anyone want to solve this type of problem?

3. Maintain an Active Mind

Research consistently demonstrates that we use and retain very little of what we are told, what we read, or what we watch. Learners learn because they are engaged in creating, processing, and interpreting experience, both real and simulated.

ISETA News-let-ter (exploring teaching alter-na-tives)
Winter 1989

A mind that wanders is normal. Most of us can be reading a very interesting novel and suddenly find we don't know what has happened the last few pages. Our reaction is to return to those pages and begin again where we need the review. Students frequently experience a similar experience when studying mathematics, but often they try to continue when they have no understanding of the preceding material. In mathematics, this is like building a two-story house and then trying to dig out the basement.

An active mind is a necessity for learning, and students need to be taught to maintain the mind's activity. "AHA!" experiences are the best learning experiences we can bring our students. We have excellent clues on how to build situations that are nurturing climates for "AHA!" but those situations are difficult to create in schools. The major reasons: Time and Accountability.

Two types of strategies should be applied on a mathematical problem. A cognitive strategy is an identifiable and reproducible thought process directed at a particular type of task (the quadratic formula for solving equations). A heuristics strategy is the use of a smorgasbord of thinking tools when reacting to a situation (solving nonroutine problems). Some students want every problem reduced to a cognitive strategy, but that is neither possible nor desirable. Wherever possible, use heuristics. The best way to do this is through the use of nonroutine problems that is, problems where no cognitive strategy has been developed or where the cognitive strategies are more difficult to apply than heuristics.

Keeping an active mind may be translated as "THINKING," but a major problem with the translation is the meaning of the word. Every teacher-tutor aspires to teach "thinking," but most descriptions of how that is accomplished are vague and difficult to replicate.

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What does this mean?

What does 3 + 5 mean?
What does "Find the least common multiple of 21 and 35" mean?
What does 7% of 832 mean?
What does x + 5 = 12 mean?
What does (x - 5)2 = 49 mean?

 

Examples of other questions which search for meaning are:

Write a simpler problem.
What does the answer to this problem look like?
If you had the answer, what would you do to check it?
Find the last problem you could do correctly.
What is different about this problem from others you have done?
If you could change something about this problem to make it easier, what would you select?

Teachers of mathematics need to primarily ask "W" questions (What, Where, Why, When, Who). They need to avoid asking "How" questions because these frequently encourage the replication of some known process.

Teacher questions are rarely as good as student questions. And the very best questions are those that a student asks of him/herself. When a student questions her/himself, it requires:
(1) active processing, (2) thinking about their own thinking processes, and (3) the recall of prior knowledge.

4. Monitor Comprehension

The skilled learner strives to reach two goals:
to understand the meaning of the tasks at hand, and
to regulate his or her own learning

Strategic Teaching and Learning, ASCD

Self-evaluation is a crucial, on-going process. Just as prior knowledge is crucial to beginning successful study, so too is review crucial for integrating and consolidating it. The students who pass one test and immediately forget its content have never learned that constant review and assimilation of knowledge is a necessity in mathematics.

Teach students to treat evaluation and review as important aspects of their study. You might begin this process with some learning theory. Learning theory should be a first course in education, but few students or teachers seem to act consciously upon one. The three step

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process shown below is sufficient for students to apply and monitor their own effectiveness in learning.

1. Awareness of the new learning intended

a. Background check (prior knowledge needed)
b. Focus (put attention directly on the new learning)

2. Active response to some question or problem intended to illustrate acquisition of the new learning.

3. Feedback on the degree to which the active response was appropriate or correct.

Students also will benefit by evaluating their instruction. Making a conscious effort to judge their instruction is part of the process for students becoming responsible for their own success/failure. Bloom's criteria for quality instruction provides a simple four-phase method for evaluation.

1. Does the instruction provide clear cues or directions?

2. Does the instruction include an appropriate learning activity?

3. Does the instruction provide feedback?

4. When difficulties are encountered, does the instruction provide corrective recycling?

At the heart of comprehension monitoring are those skills which are labeled metacognitive. Metacognitive skills are the thoughts (knowledge and skills) used to plan, monitor, and evaluate an individual's cognition. If cognition were thinking, then metacognition would be thinking about thinking. The monitoring and evaluating functions of metacognition are important when a student has only a vague idea of how well or poorly they are learning. Many students are unaware of what they know and what they do not know. These students are having trouble monitoring and evaluating their comprehension. Successful students learn efficiently by utilizing the feedback they receive from the monitoring and evaluating functions to improve their future performance.

I have found an interesting way to encourage my students to engage in more metacognition. I tell them that all learning of mathematics comprises two components. The first is learning the content of the subject. The second is learning to what degree that content has been learned. The permutations of these two components include the following:

1. The student who knows the subject and knows he/she knows the subject. This is the student who is probably working towards an "A."

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2. The student who knows the subject, but doesn't know whether he/she knows the subject. This is frequently the student with anxiety, and anxiety reduction techniques may bring a dramatic improvement in performance.

3. The student who doesn't know the subject, and also knows he/she doesn't know the subject. This student has the knowledge to improve performance. The question here is if the student has the will and management skills.

4. The student who doesn't know the subject, and doesn't know he/she doesn't know the subject. This student suffers from "double ignorance." All is not lost, however, because awareness of the situation may move this student to take advantage of the plethora of materials now available to evaluate the degree of knowledge. If so, this student can become a #3 just with that knowledge. Again, does the student have the will and management?

Students today present new, more difficult challenges for their mathematics teachers-tutors. Explicit teaching of the nature of mathematics rather than solving routine problems can best overcome the mindblocks of our students that continually lead to more failure.

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