XIII.  Meeting Academic and Workplace Standards: How this Book Can Help

Preparing for Academic Tests

Since the first publication of Problem Solving and Comprehension 20 years ago, a great deal has changed in education and in our understanding of mental processes. This chapter reviews connections between Thinking Aloud Pair Problem Solving (TAPPS), and the objectives described in recent educational framework and standard documents. Specifically, we show which framework objectives TAPPS will help you meet, and how and which parts of the major assessments can be prepared for with the Sixth Edition of Problem Solving Comprehension.

National Mathematics Standards

The NCTM Curriculum and Evaluation Standards (NCTM, 1989, pp. 5–6) describes five goals as being essential for attaining mathematical literacy:

1.   Learning to value mathematics.

2.   Becoming confident in one’s own ability.

3.   Becoming a mathematical problem solver.

4.   Learning to communicate mathematically.

5.   Learning to reason mathematically.

These are precisely the outcomes of the TAPPS process when it is applied to solving math problems.

The Standards state that “students need extensive experience listening to, reading about, writing about, speaking about, reflecting on, and demonstrating mathematical ideas. … It is not enough for students to write the answer to an exercise or even to ‘show all their steps.’ It is equally important that students be able to describe how they reach an answer or the difficulties they encountered while trying to solve a problem” (p. 140). TAPPS develops mathematical communication skills as no other instructional method can. By requiring a detailed description of each reasoning step, TAPPS builds reasoning skills while it fosters an extraordinarily deep level of mathematical communication. Yet it does this starting at the learners’ level by demanding that the problem solver and listener understand each other. This is in marked contrast to traditional approaches that demand communication according to some externally directed format (whether two-column proof or portfolio-style log book) designed to meet the needs of the teacher or some distant authority. Externally mandated structures are never as meaningful or as rigorous as those imposed by the dynamics of the problem solver-listener interaction. Because the listener is constantly monitoring and probing the problem solver’s thinking, a level of explicitness and an awareness of detail can be attained that goes far beyond that required in written proofs or other standard forms of mathematical communication.

As reasoning and communication skills improve, learners acquire a more sophisticated ability in solving problems and this in turn creates a genuine confidence based on a realistic awareness of actual accomplishments. Finally, given confidence in their skills, learners can begin to build an honest sense for the value of mathematics in their own lives.

Alignment of Whimbey and Lochhead: Problem Solving and Comprehension With the NCTM Curriculum And Evaluation Standards For School Mathematics

Alignment of Whimbey and Lochhead: Problem Solving and Comprehension With the NCTM Curriculum And Evaluation Standards For School Mathematics

National Science Standards

The National Science Education Standards (NRC, 1996) also called for an approach to instruction that was well aligned with the benefits of TAPPS. According to these Standards, “inquiry” is the key component in science learning and “inquiry requires the identification of assumptions, use of critical and logical thinking, and consideration of alternative explanations” (p. 23). These are precisely the skills that the TAPPS dynamic demands and develops.

In describing how to teach the skills of inquiry, the Standards ask teachers to “orchestrate discourse among students about science ideas” (p. 36). Without being very specific about how this can be done, the Standards outline the following:

These objectives will not happen simply because the National Academy of Science says they should. Small group discussions do not spontaneously gravitate to the consideration of how we know what we know or to a rigorous analysis of basic assumptions. Specific instructional structures are needed to encourage these outcomes and to help refine first (inevitably inadequate) efforts. The advantage that TAPPS has over other structures is that it creates a natural dynamic in which learners are responsible for these outcomes while teachers are free to monitor the big picture, to coach specific pairs who may be in need of assistance, and to insure that the overall inquiry attains its scientific objectives. By creating a learner-based dynamic to drive the search for an even greater precision of detail, TAPPS insures that inquiry becomes a learning activity rather than a teacher mandated process to which students reluctantly conform. Because Problem Solving and Comprehension provides a prerequisite preparation for the Science Standards we do not include a detailed table relating specific problems to individual standards.

National Standards for Reading and Writing

There is currently no politically acceptable consensus on national standards for reading and writing. A wide gap remains between many current educational theories and effective educational practice. This situation is particularly tragic given the enormous importance of these skills and the number of highly effective strategies that have been developed and exhaustively tested during the past two decades (e.g., Clay, 1987; Linden & Whimbey, 1990b; Whimbey, Johnson, Williams, & Linden, 1993).

Both reading and writing are highly complex skills that demand a great deal of experience and judgment. The composition or interpretation of text rarely can be accomplished through a few simple rules. That is why these skills have remained, and will continue to remain, beyond the realm of computers. Yet these complex skills depend on a number of simpler skills that can and must be fully developed. Thus, for example, the ability to compose a coherent paragraph contains, as one component, the skill of arranging a set of sentences in their most meaningful order. These skills can and should be practiced separately from the more complex activities in which they are eventually practiced.

Problem Solving and Comprehension teaches a few of the most important skills needed in reading and writing: step by step analysis, structural diagramming of relationships and sequences, and the structure and function of analogies. Additional skills are contained in The Whimbey Writing Program (Whimbey & Blanton, 1995), and in Analytical Reading and Reasoning (Whimbey, 1989).

The most important contribution of Problem Solving and Comprehension to the development of literacy is not in the specific skills it teaches, but in the strategy of TAPPS. A very similar strategy is at the heart of Reading Recovery (Clay, 1985, 1991, 1993), one of the most carefully researched education programs of all time. In this highly effective program the teacher takes on a role similar to that of the listener while the student-reader performs the role of the problem solver. Reading Recovery uses this strategy in teaching, staff training, and reading research. Another successful approach, Reciprocal Reading (Brown & Palincsar, 1989) employs a variant in which students perform roles analogous to those of the listener and problem solver. TAPPS is the least difficult strategy of the three to implement and the most adaptable to scaling up for use with large numbers of teachers and students. It is the only one of the strategies primarily designed for use in high school and beyond. It is the approach which places the greatest degree of responsibility and control in the hands of the learners. More specific descriptions of the role of TAPPS in the development of reading and writing competence can be found in Blueprint for Educational Change: Improving Reasoning, Literacies, and Science Achievement with Cooperative Learning (Whimbey et al., 1993) and in Why Johnny Can’t Write (Linden & Whimbey, 1990b).

All of the standards for reading, writing, mathematics, and science base their recommendations on the results of cognitive research into thinking and learning processes. In the next section we examine how Problem Solving and Comprehension relates to that body of research.

A Few Lessons from Cognitive Science

Modern cognitive psychology is, in many respects, a direct consequence of the digital computer. The existence of computer programs provide psychologists with a tangible model for the mental processes earlier Behaviorist Psychologists had no means for imagining. Despite cognitive psychology’s recent origins, important educational implications have been apparent for 15 years. In her Presidential address to the 1987 AERA convention, Lauren Resnick summarized some of those implications. Resnick indicated that the most effective programs for teaching thinking share three characteristics:

1.   They involve socially shared intellectual work, and they are organized around joint accomplishment of tasks, so that elements of the skill take on meaning in the context of the whole.

2.   They make usually hidden processes overt, and they encourage student observation and commentary. They also allow skill to build up bit by bit, yet permit participation even for the relatively unskilled.

3.   Most successful programs are organized around particular bodies of knowledge and interpretation—subject matter … rather than general abilities. The treatment of the subject matter is tailored to engage students in processes of meaning construction and interpretation. (Resnick, 1987)

The TAPPS process clearly fits all aspects of the first two characteristics. No other program is as effective in making usually hidden processes overt. According to Resnick’s third criterion, our application of TAPPS to test preparation could be criticized as too narrowly focused on general abilities. In fact, the most successful applications of Problem Solving and Comprehension have been those that use the text in a program focused on a specific subject matter, such as mathematics, chemistry, or English composition. We designed the book to be a short course in analytical reasoning, because we do not believe a long course would be as effective. When using Problem Solving and Comprehension keep in mind that this text was not intended to be used in isolation and that it is very important for you to extend the lessons learned to at least one specific subject matter content.

The Importance of Unimportant Contexts

Cognitive science has repeatedly revealed the importance of context. Knowledge gained in one context is not usually transferred automatically, making it useful in a second related context. From this, many educators have concluded that all skills need to be learned within a specific real-world application. In addition, the desire to make education more engaging for students encourages the use of applied problems and the avoidance of abstract or meaningless situations. Yet in this book you will find very few real-world problems. In most cases the problems deal with silly situations in which no one could ever have a really serious interest. Why is this?

This course is designed to help you improve your skills in analytical reasoning. Cognitive science has shown that emotions are often a great obstacle to clear-headed analytical thinking. When we are deeply concerned, excited, or fearful, special hormones, such as adrenaline, kick in and affect the way we think. Whenever we care strongly about the outcome of a process, our attention is focused more on some aspects of the problem than on others. We distort information to give some data special importance. These distortions can be very useful. If, for example, you are being tracked by a tiger in the jungle, it is important that you pay attention to the tiger and not to the insects that may be biting you. But distorted attention is not useful in tests or in situations where it is important to consider all the data. Controlled emotion, a passion for success, is, of course, the key to any excellent performance. But for the TAPPS program, you first need to focus your passion on attending to details, that is, on not missing any step in the process.

Learning to think aloud and to monitor your thinking carefully, in a step-by-step manner, will use every bit of your brain power. Our experience with thousands of students is that the extra distractions of important problem situations add more than our brains can handle. Cognitive science has consistently shown that the novice’s channel capacity is very limited in comparison with the expert’s. This means that learners need to focus on a few things in order to avoid getting lost in the big picture. By getting caught up in the meaning of the problem, you forget to pay attention to careful step-by-step analysis. While you are learning the first steps in TAPPS, it is best to work with simple, unimportant, silly problems.

These kinds of problems can be great fun. When they are solved using TAPPS, there is a great deal of excitement; this comes not from the problem situations but from the experience of watching the human mind at work. The thrill is similar to that in a good detective mystery. We are much less interested in what the solution is than in how the solution is found.

In sports we find a somewhat similar strategy in calisthenics. These are warm-up and training exercises designed to prepare one for playing the real game. Each exercise is simple and on its own rather silly. But practicing these exercises is critical to success on the playing field. The exercises in Problem Solving and Comprehension are designed to strengthen certain mental “muscle groups” critical to success in all areas of academic work. Without well-developed skills in these areas, you are bound to drop the ball.

Once you have mastered the exercises in this book, you will want to apply the skills you learned to real-world problem solving. But until these skills are automatic and require little conscious attention, you will find yourself forgetting to work step by step. You may get distracted by emotions, or you may just lose your chain of thought, because there are too many interesting things to keep track of. The only way to guard against these distractions is to practice the basics over and over in as simple a context as possible.

The Demands of Work

While new standards are being formulated for the academic subjects, equally important new criteria are being generated by business and labor. One extremely influential document is What Work Requires of Schools (SCANS, 1991), better known as the Secretary’s Commission on Achieving Necessary Skills (SCANS) report because it was produced by the CI.S. Department of Labor Secretary’s Commission on Achieving Necessary Skills. In it a commission made up of industry and labor leaders reports on the skills every high school graduate must have to be competitive in the work place of the 21st century. The report contains the following dramatic conclusions:

The specific skills needed are summarized in the chart that follows.

The relevance of Problem Solving and Comprehension to the skills enumerated in the SCANS report is underscored by former Secretary of Labor Robert Reich in The Work of Nations (1991): “In the vast literature on experimental learning, two works stand out: D. A. Kolb’s ‘On Management and the Learning Process,’ in D. A. Kolb et al. (Eds.), Organizational Psychology: A Book of Readings (2nd ed.; Englewood Cliffs, NJ: Prentice Hall, 1974), and A. Whimbey and J. Lockhead (sic), Problem Solving and Comprehension.”

Problem Solving and Comprehension addresses many of the SCANS competencies and foundation skills. Problem-solving strategies such as those used in problem 9 (p. 61) and problems 6 through 14 (pp. 129–131) provide an important foundation to effective use of “Resources—allocating time, money, materials, space and staff.” The listener-problem solver dialog used in TAPPS is probably the strongest foundation available on which to build “Interpersonal Skills—working on teams, teaching others, serving customers, leading, negotiating, and working well with people from culturally diverse backgrounds.” The problems in chapters 4, 9, 10, and 11 all develop important skills for “Information—acquiring and evaluating data, organizing and maintaining files, interpreting and communicating,…”

Problem Solving and Comprehension provides complete coverage of the Foundation Competencies. Working through the entire text is an excellent way to review “Basic Skills—reading, writing, arithmetic and mathematics, speaking, and listening.” Working through the complete text also establishes a strong foundation in “Thinking Skills—thinking creatively, making decisions, solving problems, seeing things in the mind’s eye, knowing how to learn, and reasoning.” Finally, the text provides a strong foundation for the important “Personal Qualities—individual responsibility, self-esteem, sociability, self-management, and integrity.” It may not be immediately apparent how the TAPPS process can build a strong basis for these personal qualities. Individual responsibility and self-esteem are qualities that each person has to construct for himself or herself. Mo amount of inspired lecturing can magically transfer these qualities from teacher to student. In TAPPS, pairs of learners have direct experience in how to be responsible for the accuracy of their own work and in understanding that they cannot always depend on an external expert or teacher for confirmation of success. This understanding is the meaning of individual responsibility and the only valid basis on which to build self esteem.

Preparing for Academic and Workplace Assessment Systems

Terra Nova

The Terra Nova (McGraw-Hill, 1997) is an example of the new student assessment systems that are being developed in response to revised national and state academic standards. Among the test’s innovations is an extensive Thinking Skills Framework. We have related this framework to the material in Problem Solving and Comprehension (PS&C I), its successor Beyond Problem Solving and Comprehension (PS&C II; Whimbey & Lochhead, 1984); The Whimbey Writing Program (WWP) and Analytical Writing and Thinking (AWT); Developing Mathematical Skills (DMS; Whimbey & Lochhead, 1981); and Analytical Reading and Reasoning (ARR) and its companion Mastering Reading Through Reasoning (MRR; Whimbey, 1995). In the following table, we show where each of these books offers instruction and practice exercises for each skill in the framework. In many cases a problem will involve more than one thinking skill and thus its placement in the table may seem somewhat arbitrary. Most of the skills appear in several parts of the Terra Nova, including both the mathematics and the English language sections. It is wise, therefore, to practice them in at least these two contexts.

Work Keys

The American College Testing service (ACT, 1995) has developed a battery of tests designed to measure essential job related skills. The Work Keys assessment package contains eight different tests each designed to measure a wide range of competencies. Tests are available in the following areas: Applied Mathematics, Applied Technology, Writing, Listening, Locating Information, Reading for Information, Teamwork, Observation.

The tests are scored on a four-level scale. The lowest level (level 3) covers skills usually learned in high school, whereas the highest levels, 6 and 7, are typical of those developed in advanced graduate degree programs. No one is expected to score high in all eight areas. ACT has established score profiles to fit a large number of different occupations. Many industries have found that the Work Keys test battery accurately measures skills that are relevant to their needs, and for this reason the test is rapidly gaining acceptance in the hiring selection process. But the test is also an excellent tool for designing a program of study. The Work Keys system is designed so that students can compare their Work Keys profile to the profile of any desired career. From this they can tell immediately which areas of competence they must develop further and by how much.

Problem Solving and Comprehension can be a useful part of any program designed to establish high school (level 3 or 4) preparation in three of the eight tests: applied mathematics, locating information, and reading for information. Similarly, Analytical Thinking and Writing provides a high school level preparation for the Writing test. Beyond Problem Soloing and Comprehension covers material that should be adequate for graduate school performance (level 7) in applied Math. We believe that a significant amount of experience with TAPPS should improve scores on the listening, observation, and teamwork tests, but at present cannot quantify that effect.