Realistic mathematics instruction as progressive mathematization

In this section we present five features which characterize realistic mathematics. At first we are dealing with learning in a context and second with the use of models. The third point (the mathematical subjects are not atomized but interwoven) is not of so much relevance for this book, while the three characteristics of the process of mathematization (construction, reflection, and interaction) are analysed in the following sections.

The new realistic approach to learning and thought process in children has far-reaching consequences. Mathematization is viewed as a constructive, interactive and reflective activity. To begin, the point of departure for education is not learning rules and formulas, but rather working with contexts. A context is a situation which appeals to children and which they can recognize in theory. This situation might be either fictional or real, and forces children to call upon the knowledge they have gained by experience − for example in the form of their own informal working methods − thereby making learning a meaningful activity for them, A well-chosen context can induce an active thought process in children, as the following example shows.

Let us start to give children of, say, 11 years the following formal and bare problem, not presented in a context: 6 ÷ 43. Many of them will have a great deal of trouble finding a solution (Streefland, 1991). Some will answer, for example: 42, 243 or 421. They manipulate at random with the given numbers, for instance   6 ÷ 3 = 2, so 6 ÷ 43 must be 42. This child views fractions as whole numbers and so do other students (Lesh et al., 1987). But some students will calculate that 6 × 4 = 24 and that 24 divided by 3 equals 8. It is true that the latter answer is correct, but when these children are questioned more closely, it turns out that they understand almost nothing about the operation which they themselves have just performed. They just remembered a rule they learned by heart, they know that the given solution is correct however they don’t know why.

Now, the same children are next given the following context problem which is accompanied by a picture: a patio is 6 metres long; you want to put down new bricks and the bricks you are going to use measure 75 centimetres in length (43 of a metre). How many bricks will you need for the length? This problem is the same as the previous one, but it has now been presented within a context, a picture of a patio and the bricks to put down. This presentation elicits a child’s own, informal approach: measuring out. This approach provides insight into the problem, something which the symbolic form (6 ÷ 43) did not do. Some students even manipulated and took the measure in reality, this means they measured out step by step 75 centimetres and after 8 steps they counted 6 metres. So the answer must be ‘eight’, they concluded. This example demonstrates that working with contexts − which, if carefully constructed, can be considered paradigmatic examples − form the basis for subsequent abstractions and for conceptualization. That is because thinking must achieve a higher, abstract level and at that level these particular contexts no longer serve a purpose. That is not to say that a process of decontextualisation occurs, but rather recontextualisation. The children continue to work with contexts, but these contexts become increasingly formal in nature; they become mathematical contexts. Their connection with the original context, however, remains clear. The process by which mathematical thinking becomes increasingly formal is called the process of progressive mathematization. Contexts, thus, have various functions. They may refer to all kind of situations and to fantasy situations (Van den Heuvel-Panhuizen, 1996). It is important that the context offer support for motivation as well as reflection. A context should indicate certain relevant actions (to take measures in the example above), provide information which can be used to find a solution-strategy and/or a thinking-model.

Of course, leaving the construction to the students does not guarantee the development of successful strategies. However it guarantees that students get the opportunity to practice mathematician’s thinking and problem solving processes. Strategies are tried, tested and elaborated in various situations.

In the previous discussion we have not argued that a student presented with ‘bare’ numerical tasks (like 6 ÷ 43) will necessarily fail to solve the problem. Hence we were not suggesting either that students who are given context problems will necessarily produce the right solution. In recent research there is found a strong tendency of children to react to context problems (‘word problems’) with disregard for the reality of the situations of these problems. Let us give two examples of items used in research (Greer, 1997; Verschaffel et al., 1997):

− ‘An athlete’s best time to run a mile is 4 minutes and 7 seconds. About how long would it take him to run 3 miles?’

− ‘Steve has bought 4 planks of 2.5 metre each. How many planks of 1 metre can he get out of these planks?’

In four studies, discussed by Greer (1997), the percentage of the number of students demonstrating any indication of taking account of realistic constraints is: 6%, 2%, 0% and 3%. The student’s predominating tendency to apply rules clearly formed an impediment to thoroughly understanding the situation.

Verschaffel et al. (1997) confronted a group of 332 students (teachers in training) with word problems and found they produced ‘realistic’ responses in only 48% of cases. Moreover the pre-service teachers considered these ‘complex and tricky word problems’ as inappropriate for (fifth grade) children. The goal of teaching word problem solving in elementary school, after their opinion, was “...learning to find the correct numerical answer to such a problem by perforn1ing the formal-arithmetic operation(s) ‘hidden’ in the problem” (Verschaffel et al., 1997, p. 357).

When solving word problems students should go beyond rote learning and mechanical exercises to apply their knowledge (Wyndhamn & Säljö, 1997). Their research showed that students (10-12 years of age) gave in most cases logically inconsistent answers. The authors interprete these findings by claiming that the students focus on the syntax of the problem rather than on the meaning. That means that the well-known rule-based relationship between symbols results in less of attention being paid to the meaning. The students follow another ‘rationality’, that is, they consider word problems as mathematical exercises “… in which a algorithm is hidden and is supposed to be identified.” (Wyndhamm & Säljö, p. 366). Hence they do not know or realize that they are expected to solve a real life problem.

Reusser and Stebler (1997) discuss another interesting research finding namely the fact that pupils ‘solved’ unsolvable problems without ‘realistic reactions’. For example:

− ‘There are 125 sheep and 5 dogs in a flock. How old is the shephard?’ (Greer, 1997).

A pupil questioned by the investigators gave as his opinion: ‘It would never have crossed my mind to ask whether this task can be solved at all’. And another pupil said: ‘Mathematical tasks can always be solved’. One of the author’s conclusions is that a change is needed from stereotyped and semantically poor, disguised equations to the design of intellectually more challenging ‘thinking stories’. What we need are better problems and better contexts. Finally, Reusser and Stebler (1997) − following Gravemeijer (1997) − give as their interpretation of the research findings that the children are acting in accordance with a typical school mathematics classroom culture.

Second, the process of mathematization is characterized by the use of models. Some examples are schemata, tables, diagrams, and visualizations. Searching for models − initially simple ones − and working with them produces the first abstractions. Children furthermore learn to apply reduction and schematization, leading to a higher level of formalization. We will demonstrate, once again this using the previous example. To begin, children are able to solve the brick problem by manipulating concrete materials. For instance, they might attempt to see how often a strip of paper measuring 43 of a metre fits in a 6-metre-long space. At the schematic level, they visualize the 6-metre-long patio and draw lines which mark out each 43 of a metre or 75 centimetres. The child adds 75 + 75 + 75... until the 6 metres have been filled The visualization looks as follows:

An example of reasoning on a formal-symbolic level is as follows: 75 centimetres fits into 3 metres 4 times. We have 6 metres, so we need 2 × 4 = 8 bricks. The formula initially tested can also be applied, but this time with insight: 41 metre fits 4 times into 1 metre, so it fits 24 times into 6 metres. But I only have 43 of a metre, so I have to divide 24 by 3, and that makes 8. At this formal level, moreover, the teacher can also explore the advantages and disadvantages of the two methods with the children.

Third, an important element of realistic mathematics instruction is that subjects and curricula (such as fractions, measurement and proportion) are interwoven and connected, whereas in the past, the subject matter was divided − and so atomized. Fourth, two other important characteristics of the process of mathematization are that it is brought about both by a child’s own constructive action and by the child’s reflections upon this action. Finally, learning mathematics is not an individual, solitary activity, but rather an interactive one.

One of the most enduring ideas concerning mathematics instruction is the following: mathematics consists of a set of indisputable rules and knowledge; this knowledge has a fixed structure and can be acquired by frequent repetition and memorization. In the past twenty-five years, far-reaching changes have taken place in mathematics instruction. More than in any other field, such changes were influenced by mathematicians who had come to view their discipline in a different light. Their observations went a long way towards stimulating a process of renewal in mathematics instruction. New consideration was given to such fundamental questions as: how might mathematics best be taught, how might children be encouraged to show more interest for mathematics, how do children actually learn mathematics, and what is the value of mathematics?

According to Goffree, Freudenthal, and Schoemaker (1981), the subject of mathematics is itself an essential element in ‘thinking’ through didactical considerations in mathematics instruction. Moreover, the notion is emphasized that knowledge is the result of a learner’s activity and efforts, rather than of the more or less passive reception of information. Mathematics is learned, so to say, on one’s own authority. From a teacher’s point of view there is a sharp distinction made between teaching and training. To know mathematics is to know why one operates in specific ways and not in others. This view on mathematics education is the basic philosophy in this chapter (Von Glazersfeld, 1991) In order to understand current trends in mathematics education, we must consider briefly the changing views on this subject.

The philosophy of science distinguishes three theories of knowledge. Confrey (1981) calls these absolutism, progressive absolutism and conceptual change. In absolutism, the growth of knowledge is seen as an accumulation, a cumulation of objective and empirically determined factual material. According to progressive absolutism a new theory may correct, absorb, and even surpass an older one. Proponents of the idea of conceptual change have defended the point of view that the growth of knowledge is characterized by fundamental (paradigmatical) changes and not by the attempt to discover absolute truths. One theory may have greater force and present a more powerful argument than another, but there are no objective, ultimate criteria for deciding that one theory is incontrovertibly more valid than another (Lakatos, 1976). Mathematics has long been considered an absolutist science. According to Confrey (1981), it is seen as the epitome of certainty, immutable truths and irrefutable methods. Once gained, mathematical knowledge lasts unto eternity; it is discovered by bright scholars who never seem to disagree, and once discovered, becomes part of the existing knowledge base.

Leading mathematicians however have now abandoned the static and absolutist theory of mathematics (Whitney, 1985). Russell (in Bishop, 1988) once explained that mathematics is the subject in which we never know what we are talking about, nor whether what we are saying is true. Today mathematics is more likely to be seen as a fluctuating product of human activity and not as a type of finished structure (Freudenthal, 1983). Mathematics instruction should reveal how historical discoveries were made. It was not (and indeed is still not) the case that the practice of mathematics consists of detecting an existing system, but rather of creating and discovering new ones. This evolving theory of mathematics also led to new ideas concerning mathematics instruction. If the essence of mathematics were irrefutable knowledge and ready-made procedures, then the primary goal of education would naturally be that children mastered this knowledge and these procedures as thoroughly as possible. In this view, the practice of mathematics consists merely of carefully and correctly applying the acquired knowledge If, however, mathematicians are seen as investigators and detectives, who analyse their own and others’ work critically, who formulate hypotheses, and who are human and therefore fallible, then mathematics instruction is placed in an entirely different light. Mathematics instruction means more than acquainting children with mathematical content, but also teaching them how mathematicians work, which methods they use and how they think. For this reason, children are allowed to think for themselves and perform their own detective work, are allowed to make errors because they can learn by their mistakes, are allowed to develop their own approach, and learn how to defend it but also to improve it whenever necessary. This all means that students learn to think about their own mathematical thinking, their strategies, their mental operations and their solutions.

Mathematics is often seen as a school subject concerned exclusively with abstract and formal knowledge. According to this view, mathematical abstractions must be taught by making them more concrete. This view has been opposed by Freudenthal (1983) among others. In his opinion, we discover mathematics by observing the concrete phenomena all around us. That is why we should base teaching on the concrete phenomena in a world familiar to children. These phenomena require the use of certain classification techniques, such as diagrams and models (for example, the number line or the abacus). We should therefore avoid confronting children with formal mathematical formulas which will only serve to discourage them, but rather base instruction on rich mathematical structures, as Freudenthal calls them, which the child will be able to recognize from its own environment. In this way mathematics becomes meaningful for children and also makes clear that children learn mathematics not by training formulas but by reflecting on their own experiences.

In the 1970s, the new view of mathematics, often referred to as mathematics as human activity, led to the rise of a new theory of mathematics instruction, usually given the designation: realistic. As it now appears, this theory is promising, but it is not the only theoretical approach in mathematics instruction; three others can be distinguished: the mechanistic, the structuralist and the empirical (Treffers, 1991). 

− The mechanistic approach reflects many of the principles of the behaviouristic theory of learning; the use of repetition, exercises, mnemonics, and association comes to mind. The teacher plays a strong, central role and interaction is not seen as an essential element of the learning process. On the contrary, mathematics class focuses on conclusive standard procedure.

− According to the second approach − the structuralist − thinking is not based on the children’s experiences or on contexts, but rather on given mathematical structures. The structuralist tends to emphasize strongly the teacher’s role in the process of learning.

− The outstanding feature of the third trend − the empirical − is the idea that instruction should relate to a child’s experiences and interests. Instruction must be child-oriented. Empiricists believe that environmental factors form the most important impetus for cognitive development (Papert, 1980). Empiricists emphasize spontaneous actions.

Critical thinking in mathematics education

The intellectual roots of critical thinking are as ancient as its etymology, traceable, ultimately, to the teaching practice and vision of Socrates 2500 years ago. His method of questioning is now known as the “Socratic questioning” and is the best known critical thinking teaching strategy. Socrates practice was followed by the critical thinking of Plato, Aristotel and the Greek skeptics, all of whom emphasized that things are often very different from what they appear to be and only the trained mind is prepared to see through the way things look to us on the surface to the way they really are beneath the surface. In the middle ages, the tradition of systematic critical thinking was embodied in the writings and teachings of thinkers such as Thomas Aquinas. During the Renaissance, a flood of scholars in Europe began to think critically about religion, art, society, human nature, law and freedom. Among these scholars were Colet, Erasmus, More, Bacon. Fifty years later in France, Descartes wrote the Rules for the Direction of the Mind, where he argued for the need for a special systematic disciplining of the mind to guide in thinking, so that every part of thinking should be questioned, doubted and tested. The critical thinking of the Rennaissance and post Rennaissace scholars opened the way for the emergence of science and for the development of democracy, human rights and freedom of thought.

It was in the spirit of intellectual freedom and critical thought that people such as Robert Boyle and Isaac Newton did their work. In his Sceptical Chymist, Boyle severely criticized the chemical theory that preceded

him. Newton, in turn, developed a far-reaching framework of thought which roundly criticized the traditionally accepted world view. Another significant contribution to critical thinking was made by the thinkers of the French Enlightenment: Bayle, Montesquieu, Voltaire and Diderot. They all began with the premise that the human mind, when disciplined by reason, is better able to figure out the nature of the social and political world.

In the 19th Century, critical thought was extended even further into the domain of human social life by Comte and Spencer. In the 20th Century, our understanding of the power and nature of critical thinking has emerged in increasingly more explicit formulations. To sum up, the tools and resources of the critical thinker have been vastly increased in virtue of the history of critical thought. Hundreds of thinkers have contributed to its development. Each major discipline has made some contribution to critical thought. Yet most educational purposes, it is the summing up of base-line common denominators for critical thinking that is most important. Let us consider now that summation. The result of the collective contribution of the history of critical thought is that the basic questions of Socrates can now be much more powerfully and focally framed and used. In every domain of human thought, and within every use of reasoning within any domain, it is now possible to question:

1. ends and objectives,

2. the status and wording of questions,

3. the sources of information and fact,

4.the method and quality of information collection,

5.the mode of judgement and reasoning used,

6. the concepts that make that reasoning possible,

7.the assumptions that underlie concepts in use,

8.the implications that follow from their use, and

9. the point of view of the frame of reference within which reasoning takes place.

In other words, questioning that focuses on these fundamentals of thought and reasoning are now baseline in critical thinking. It is beyond question that intellectual errors or mistakes can occur in any of these dimensions, and that students need to be fluent in talking about these structures and standards.

In the field of education, it is generally agreed upon that Critical Thinking capabilities are crucial to one’s success in the modern world, where making rational decisions is increasingly becoming a part of everyday life. Students must learn to test reliability, raise doubts, investigate situations and alternatives, both in school and in everyday life. As will be discussed, as well as acquiring CT, it is important to assess students’  application of their CT in different contexts. Many studies investigate CT in general, or in fields other than Mathematics, but few discuss CT in Mathematics. This study will explore CT in the context of a probability session. This research is based on three key elements: (a) Ennis’ CT taxonomy that includes CT skills ( Ennis, 1989), (b)The Learning unit "probability in daily life" (Liberman & Tversky 2002). (c) The Infusion approach between subject matter and thinking skills (Swartz, 1992).

Ennis defines CT as “reasonable reflective thinking focused on deciding what to believe or do.” In light of this definition, he develops a CT taxonomy that relates to skills that includes not only the intellectual aspect but the behavioral aspect as well. In addition, Ennis's taxonomy includes skills, dispositions and abilities (1989). The details of this alignment follow: Dispositions towards CT – A defined search for a thesis, questions and explanations, being sufficiently informed, using reliable sources, taking the overall situation into account, being relevant to the main issue, looking for alternatives, seriously considering other peoples' point of view, the suspension of judgment, taking a stand, striving for accuracy, dealing with the components of an issue in an orderly fashion, and sensitivity. Abilities in CT – focusing on the question, analyzing arguments, raising questions, evaluating the source's reliability, deduction, induction, value judgments, concept definition, assumption identification, taking actions, and interacting with others. Ennis claims that CT is a reflective (by critically thinking, one’s own thinking activity is examined) and practical activity aiming for a moderate action or belief. There are five key concepts and characteristics defining CT according to Ennis: practical, reflective, moderate, belief and action.

Promoting critical thinking and problem solving in mathematics education is crucial in the development of successful students. Critical thinking and problem solving go hand in hand. In order to learn mathematics through problem solving, the students must also learn how to think critically. There are five values of teaching through problem solving:

1. problem solving focuses the student;s attention on ideas and sense making rather than memorization of facts;

2. problem solving develops the student0s belief that they are capable of doing mathematics and that mathematics makes sense;

3.it provides ongoing assessment data that can be used to make instructional decisions, help students succeed, and inform parents;

4. teaching through problem solving is fun and when learning is fun, students have a better chance of remembering it later.

Some principles of problem solving:

The primary objective is to help the student to become aware of the fact that problem solving is not a special area but instead uses the same logical processes to which they are already familiar and use routinely. The problem statement itself is the primary cause of novice students difficulty in solving word problems. The solution is to ignore, when reading a problem statement, any phrases that start with words like “if”. The initial action in starting a solution is identifying what is asked for. The student must be learned to verbalize. A verbal statement following the final result is of particular importance: what does the result tell me? In addition to completing the solution, the ending statement serves as a quick check of one;s work. An adequate solution presentation does not have to be explained.

There are two main approaches to fostering CT: the general skills approach which is characterized by designing special courses for instructing CT skills, and the infusion approach which is characterized by providing these skills through teaching the set learning material. According to Swartz, the Infusion approach aims for specific instruction of special CT skills during the course of different subjects. According to this approach there is a need to reprocess the set material in order to combine it with thinking skills.

This report is a description of an initial study, a snap shot that focused on one session and demonstrates the entire study. In this report, we will show how the mathematical content of "probability in daily life” was combined with CT skills from Ennis' taxonomy, reprocessed the curriculum, tested different learning units and evaluated the subjects' CT skills. Moreover, one of the overall research purposes is to examine the effect of the Infusion approach on the development of critical thinking skills through probability sessions. The comprehensive research purpose will be to examine the effect of learning by the Infusion approach using the Cornell questioners (a quantitative test) and quantitative means.

Mathematics is often held up as the model of a discipline based on rational thought, clear, concise language and attention to the assumption and decision-making techniques that are used to draw conclusions. In 1938, Harold Fawcett introduced the idea that students could learn mathematics through experiences of critical thinking. His goals included the following ways that students could demonstrate that they were, in fact, thinking critically, as they participated in the experiences of the classroom:

1. Selecting the significant words and phrases in any statement that is important, and asking that they be carefully defined.

2. Requiring evidence to support conclusions they are pressed to accept.

3. Analyzing that evidence and distinguishing fact from assumption.

4. Recognizing stated and unstated assumptions essential to the conclusion.

5. Evaluating these assumptions, accepting some and rejecting others.

6. Evaluating the argument, accepting or rejecting the conclusion.

7. Constantly reexamining the assumptions that are behind their beliefs and actions.

The critical thinking is still present in the goals, but it has been subsumed by more holistic notions of what it means to teach, do and understand mathematics. The students will be able to:

1. Organize and consolidate their mathematical thinking through communication;

2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others;

3. Analyze and evaluate the mathematical thinking and strategies of others;

4. Use the language of mathematics to express mathematical ideas precisely.

These ideas are very similar to those promoted by Fawcett in 1938. Little has changed in the mainstream ways that people tend to define critical thinking in the context of mathematics education. Students are expected to search for the strengths and weaknesses of each and every strategy offered. It is no longer good enough to reach an answer to a problem that was posed. Now, students are cajoled into communicating their own ideas well, and to demand the same communication from others. A shift has occured from listing skills to be learned toward attributes of classrooms that promote critical thinking as part of the experience of that classroom. Such a class to promote critical thinking can be created by providing the conditions for the students to communicate with one another in order to reflect together on the solution to the problem. The first condition is for the students to feel free in expressing their ideas. Then, they must be able to listen attentively to their classmates and show interest in their ideas. So, they communicate both for learning mathematics and in mathematical terms. On the other hand, the students get accustomed to group work which implies mutual help and cooperation for a mutual aim.

The Problem With Problem Solving

The main problem with problem solving lies in the fourth element listed above: problem solving is a heuristic. Recall that a heuristic is a guideline that may or may not yield success but, unlike an algorithm, it does not depend on knowledge of the problem to be successful. Heuristic is a method of thought that does not pertain to any specific problems or content. The element is problematic because it contradicts three other elements within the theory: the definition of problem solving, successful problem solving requires a knowledge base, and problem solving enables learners to transfer knowledge. Each of these three elements implies that previously learned knowledge of the problem is necessary to solving the problem, whereas use of a heuristic assumes no knowledge is necessary.

I argue, like Peikoff (1985), that there is no way to separate thinking or problem solving from knowledge. Just like instruction and curriculum, these concepts imply one another and cannot be discussed separately for long. Likewise, to acquire knowledge, one must think. This is not to say that students cannot construct knowledge as they solve a given problem, only to say that often the problems they are presented only require

them to apply existing knowledge. From this perspective, it must be assumed that students do not construct all of the knowledge in a given curriculum. Yet problem solving as a heuristic is the most cherished aspect of problem solving because it is content-less. For example, in the preface to Mathematical Discovery, George Polya (1962), one of the foremost thinkers on problem solving says, I wish to call heuristic the study that the present work attempts, the study of means and methods of problem solving. The term heuristic, which was used by some philosophers in the past, is half forgotten and half-discredited nowadays, but I am not afraid to use it.

In fact, most of the time the present work offers a down-to-earth practical aspect of heuristic. (p. vi) Instructional textbooks sometimes play off this process versus content dichotomy: a teacher can either teach students to be critical thinkers and problem solvers or she can teach students more content knowledge. The authors of one textbook say, Too often children are taught in school as though the answers to all the important questions were in textbooks. In reality, most of the problems faced by individuals have no easy answers. There are no reference books in which one can find the solution to life’s perplexing problems. (Gunter, Estes, & Schwab, 2003, pp. 128–129) The dichotomy implies that thinking and knowledge are mutually exclusive, when in fact critical thinking and problem solving require a great deal of specific content knowledge. 

Problem solving and heuristics cannot be contentless and still be effective. Critical thinking, problem solving, and heuristics must include a knowledge base (Fredricksen, 1984; Ormrod, 1999). Including the knowledge base enables the principle cognitive function of problem solving—the application of conceptual knowledge, or transfer—to occur (Peikoff, 1985). However, the degree to which Dewey and Polya actually believed that a heuristic could be completely content-less and still be effective is not clear. Further, many instructional textbooks actually stress the importance of content knowledge in solving problems (Henson, 2004; Kauchak & Eggen, 2007; Lang & Evans, 2006).

Each of the above elements of problem solving will be reviewed again in light of the relationship between thinking and knowledge and the research base on problem solving. Element one, the definition of a problem, implies that one must have some knowledge of the problem to solve it. How can one solve a problem without first knowing what the problem is? In fact, identification of the problem is what is called for in the first two steps, Read and Explore, of the heuristic. In this step, the student first becomes aware of the problem and then seeks to define what it is or what the problem requires for its solution. Awareness and definition comprise the knowledge that is essential to solving the problem. Consider the effectiveness of students relative to their respective experiences with a given problem. The student more familiar with the problem will probably be better able to solve it. In contrast, the student new to the problem, who has only studied the heuristic, would have to re-invent the solution to the problem.

So the first two steps of the heuristic imply that one needs a great deal of knowledge about the problem to be an effective problem solver. In fact, if one wants to solve the problem for the long term, one would want to thoroughly study the problem until some kind of principles were developed with regard to it. The final outcome of such an inquiry, ironically, would yield the construction of an algorithm.

The second element, the definition of problem solving, also implies a connection between thinking and knowledge. It says that problem solving is essentially applying old knowledge to a new situation (Krulik & Rudnick, 1987). However, if knowledge or a problem is genuinely new, then the old knowledge would not apply to it in any way. Ormrod (1999) suggests that the so-called new situation is really the same as the old in principle. For example, the principle of addition a student would use to solve the problem  1+ 2 = 3 is essentially the same principle one would apply to 1 + x = 3. The form may be different but ultimately the same principle is used to solve both problems. If this is the case, then a more proper element of problem solving would be number eight, the transfer of knowledge or application of conceptual knowledge.

The third and fourth elements algorithms and heuristics are problematic. Krulik and Rudnick (1980) distinguish between algorithms and heuristics. Unlike employing an algorithm, using a heuristic requires the problem solver to think on the highest level and fully understand the problem. Krulik and Rudnick also prefer heuristics to algorithms because the latter only applies to specific situations, whereas a heuristic applies to many as yet undiscovered problems. However, an algorithm requires more than mere memorization; it requires deep thinking too. First, in order to apply an algorithm, the student must have sufficient information about the problem to know which algorithm to apply. This would only be possible if the student possessed a conceptual understanding of the subject matter. Further, even if a student could somehow memorize when to apply certain algorithms, it does not follow that he or she would also be able to memorize how to apply it (Hu, 2006; Hundhausen & Brown, 2008; Johanning, 2006; Rusch, 2005).

Second, algorithms and problem solving are related to one another. Algorithms are the product of successful problem solving and to be a successful problem solver one often must have knowledge of algorithms (Hu, 2006; Hundhausen & Brown, 2008; Johanning, 2006; Rusch, 2005). Algorithms exist to eliminate needless thought, and in this sense, they actually are the end product of heuristics. The necessity to teach heuristics exists, but heuristics and algorithms should not be divided and set against one another. Rather, teachers should explain their relationship and how both are used in solving problems.

A secondary problem that results from this flawed dichotomy between algorithms and heuristics is that advocates of problem solving prefer heuristics because algorithms only apply to specific situations, whereas heuristics do not pertain to any specific knowledge. If one reflects upon the steps of problem solving listed above one will see that they require one to know the problem to be successful at solving it. Consider the sample problem above to which the heuristic was applied. If one knows the heuristic process and possesses no background knowledge of  similar problems, one would not be able to solve the problem. For example, in the first step of the heuristic one is supposed to Read the problem, identify the problem, and list key facts of the problem. Without a great deal of specific content knowledge how will the student know what the teacher means by “problem,” “key facts,” and so on? The teacher will probably have to engage the student in several problems. Without extensive knowledge of facts, how does the student know what mathematical facts are, and how they apply to word problems, for example?

In the second step, Explore, the problem solver looks for a pattern or identifies the principle or concept. Again, how can one identify the pattern, principle, or concept without already possessing several stored patterns, principles, and concepts? Indeed, to a student with very little mathematical knowledge, this problem would be extremely difficult to solve. The heuristic would be of little help. The heuristic says to draw a diagram, presumably to make the problem more concrete and therefore more accessible to the student, but without already knowing what the concept the problem exhibits this would very difficult, if not impossible. Using the chart with the data as an example, it would require previous knowledge in mathematics to be able to construct it. It seems that the heuristic in this problem is in reality just another algorithm that the teacher will have to teach as directly and as repetitively until the students learn how and when to apply it, which is the very opposite of what advocates of problem solving want. The same is also true of step five, Review and Extend. Presumably if a student could represent this problem in algebraic form, he or she should also be able to solve the same problem without recourse to drawing diagrams,  recording data, etc. One could simply solve the problem right after step one.

The sample problem illustrates what scientists have discovered about novices and experts. In studies that examined expert and novice chess players, researchers found that their respective memories were no different in relation to random arrangements of chess pieces. When the pieces were arranged in ways that would make sense in a chess game, the experts’ memories were much better. The theory is that an expert chess player is not a better problem solver, he or she just has a more extensive knowledge base than a novice player. He or she is past the rudimentary hypothesis testing stage of learning, past the problem solving heuristic stage and is now simply applying algorithms to already-solved problems (Ross, 2006). The same could be said for students applying a heuristic to the above problem. The only ones who could solve it would be those who use an algorithm. Even if a teacher taught the heuristic to students, he or she would essentially be teaching an algorithm.

Advocates of problem solving are not solely to blame for the misconception between thinking and knowledge and between heuristics and algorithms. The misconception is likely due to teachers that have overused algorithms and never shown students how they are formed, that they come from heuristics, and that

one should have a conceptual understanding of when they should be used, not merely a memorized understanding of them. The fundamentally flawed dichotomy within problem solving probably stems from thinking in terms of “either-ors.” One side defines appropriate education as teaching algorithms by having students memorize when to use them but not why. The other side, by contrast, emphasizes that thinking for understanding is preferable to simply memorized knowledge. Perhaps what has happened in the shift from the former to the latter practices is the instructional emphasis has shifted from content to thinking so much that the knowledge base has been wiped out in the process. Ironically, eliminating knowledge from the equation also eliminates the effectiveness of problem solving.

The dichotomy between knowledge and thinking has also affected elements five and six. Number five states that problem solving connects theory and practice. At the core of this element is yet another flawed dichotomy. Many educators hold that education should prepare students for the real world by focusing less on theory and more on practice. However, dividing the two into separate cognitive domains that are mutually exclusive is not possible. Thinking is actually the integration of theory and practice, the abstract and the concrete, the conceptual and the particular.

Theories are actually only general principles based on several practical instances. Likewise, abstract concepts are only general ideas based on several concrete particulars. Dividing the two is not possible because each implies the other (Lang & Evans, 2006). Effective instruction combines both theory and practice in specific ways. When effective teachers introduce a new concept, they first present a perceptual, concrete example of it to the student. By presenting several concrete examples to the student, the concept is better understood because this is in fact the sequence of how humans form concepts (Bruner, Goodnow, & Austin, 1956; Cone 1969; Ormrod, 1999; Peikoff, 1993). They begin with two or more concrete particulars and abstract from them the essential  defining characteristics into a concept. For example, after experiencing several actual tables a human eventually abstracts the concept a piece of furniture with legs and a top (Lang & Evans, 2006).

On the other hand, learning is not complete if one can only match the concept with the particular example of it that the teacher has supplied. A successful student is one who can match the concept to the as yet unseen examples or present an example that the teacher has not presented. Using the table as an example, the  student would be able to generate an example of a new table that the teacher has not exhibited or discussed.

This is an example of principle eight, the transfer of knowledge or applying conceptual knowledge. The dichotomy between theory and practice also seems to stem from the dichotomous relationship between the teaching for content-knowledge and teaching for thinking. The former is typically characterized as teaching concepts out of context, without a particular concrete example to experience through the five senses. The latter, however, is often characterized as being too concrete. Effective instruction integrates both the concrete and abstract but in a specific sequence. First, new learning requires specific real problems. Second, from these concrete problems, the learner forms an abstract principle or concept. Finally, the student then attempts to apply that conceptual knowledge to a new, never before experienced problem (Bruner, Goodnow, & Austin, 1956; Cone, 1969; Ormrod, 1999; Peikoff, 1993).

The theory vs. practice debate is related to problem solving because problem solving is often marketed as the integration of theory and practice. I argue, however, it leaves out too much theory in its effort to be practical. That is, it leaves out the application of conceptual knowledge and its requisite knowledge base. Element six, problem solving teaches creativity, is also problematic. To create is to generate the new, so one must ask how someone can teach another to generate something new. Are there specific processes within a human mind that lead to creative output that can also be taught? The answer would depend at least in part on the definition of create. When an artist creates, he or she is actually re-creating reality according to his or her philosophical viewpoint, but much, if not all, of what is included in the creation is not a creation at all but an integration or an arranging of already existing things or ideas. So in one sense, no one creates; one only integrates or applies previously learned knowledge. No idea is entirely new; it relates to other ideas or things. The theory of relativity, for example, changed the foundational assumptions of physics, but it was developed in concert with ideas that already existed. There may be no such thing as pure creativity, making something from nothing. What seems like creativity is more properly transfer or the application of concepts, recognizing that what appears like two different things are really the same thing in principle.

On the other hand, it is possible to provide an environment that is conducive to creativity. Many problem solving theorists have argued correctly for the inclusion of such an atmosphere in classrooms (Christy & Lima, 2007; Krulik & Rudnick, 1980; Slavin, 1997; Sriraman, 2001). I only object to the claim that problem solving teaches creativity defined as creating the new. It can, however, teach creativity defined as the application of previously learned principles to new situations.

Element seven, problem solving requires a knowledge base, although not problematic is only neglected within the theory of problem solving. This is ironic given how important it is. Jeanne Ormrod (1999) says, “Successful (expert) problem solvers have a more complete and better organized knowledge base for the problems they solve” (p. 370). She also relates how one research inquiry that studied the practice of problem solving in a high school physics class observed that the high achievers had “better organized information about concepts related to electricity” (p. 370). Not only was it better organized, the students were also aware of “the particular relationships that different concepts had with one another” (Cochran, 1988, p. 101). Norman (1980) also says, I do not believe we yet know enough to make strong statements about what ought to be or ought not to be included in a course on general problem solving methods. Although there are some general methods that could be of use…I suspect that in most real situations it is…specific knowledge that is most important. (p. 101)

Finally, element eight, problem solving is the application of concepts or transfer, is also not problematic; it too is only neglected within the theory of problem solving. Norman Frederiksen (1984) says, for example, “the ability to formulate abstract concepts is an ability that underlies the acquisition of knowledge. [Teaching how to conceptualize] accounts for generality or transfer to new situations” (p. 379).   According to this passage, it is the application of conceptual knowledge and not the heuristic alone that as Frederiksen says, “accounts for generality or transfer,” (p. 379) which the advocates of problem solving so desire.

Problem solving would be more effective if the knowledge base and the application of that knowledge were the primary principles of the theory and practice. Currently, it seems that a content-less heuristic is the primary principle, which, as I have argued, is problematic because it dichotomizes thinking and knowledge into two mutually exclusive domains. In fact, in the course of solving any problem one will find themselves learning of all things not a heuristic, but an algorithm. In other words, teachers must not only teach students the heuristic and set their students free upon the problems of everyday life. Rather, teachers must, in addition to teaching students sound thinking skills, teach them what knowledge in the past has been successful at solving the problems and why.

Pedagogical and cognitive aspects of problem solving

As modern civilization requires relentless quantification and critical evaluation of information in daily transactions, it becomes necessary to develop newer ways of thinking and reasoning that can be used to learn and do mathematical activities. Through problem solving for instance, we acquire a functional understanding of mathematics needed to cope with the demands of society.

School mathematics of the twenty first century is viewed by educators to be that which should engage a learner in problem solving and reasoning. It should also foster deep understanding and develop the learner’s critical and analytical thinking. Instruction should not be limited to plain mastery of algorithms or the development of certain mathematical skills. It should involve learners in investigation through “exploring, conjecturing, examining and testing” (NCTM, 1990, p.95). It should be tailored to promote reflective thinking among students. A wealth of research on mathematics education and cognitive science in the last decade has dealt with the pedagogical and cognitive aspects of problem solving. Rivera and Nebres (1998) note specifically “the numerous published research studies of Fennema and Carpenter on Cognitively Guided

Instruction (CGI) in the last quarter of this century [which] point to the pernicious effects of status quo ways of thinking about mathematics and problem solving (i.e. existing mathematics culture)”. CGI recognizes the “acculturation of school children to an algorithmic approach to learning basic arithmetical facts” which pervade the current school mathematics culture and which have been proven to be “detrimental to children’s own ways of thinking about problem solving and computations”.

Bishop (1999) adds that “research has shown the importance of the idea of situated cognition which describes the fact that when you learn anything you learn it in a certain situation” . Thus for learning to become meaningful, the learner has to actively participate in the formation of mathematical concepts. She should not passively receive knowledge from an authority but should be involved in the construction of knowledge. The theory of active construction of knowledge influenced many learning theories formulated by staunch contemporary mathematics educators like Von Glasersfeld, Cobb, Bauersfeld, Vygotsky and numerous others (Rivera, 1999). In fact, “problem solving and mathematical investigations based on a constructivist theory of learning, have been the main innovations or revivals for the last decade” according to Southwell (1999, p.331).

Willoughby (1990) believes that the abundant books, pamphlets and courses on critical thinking and problem solving that have been propagated in the 1980s cannot be of help unless certain pedagogical misconceptions are clarified. This includes prescribed rules such as finding key words in a problem to decide the appropriate operations on the values given in the problem, or applying arithmetic algorithm to any word problem. Developing critical and analytical thinking through problem solving takes time and a lot of teacher’s commitment and dedication. (Willoughby, 1990; Barb and Quinn, 1997). Developing critical and analytical thinking involves pedagogical conceptions with a philosophical basis. This paper adheres to the constructivist theory of learning and promotes the belief that problem solving processes rest on basic thinking skills which are best developed within a constructivist framework. Another challenge of the new millenium is the proper use of the ever advancing technology in education. Researchers have to look into the quality of instruction and curriculum which utilize technology. Educational technology should be guided by pedagogical principles that guarantee effective  learning, and not subordinated to technological ends. Thus, “technology should be used to advance educational programs, [and] should not determine programs” (Witt, 1968, p. 145). How to empower students further in learning with the use of technology should be the concern of curriculum designers. In the light of existing literature base on mathematics instruction and flourishing research studies on mathematics teaching and learning, this paper explores issues and finds ways of fostering critical and analytical thinking through problem solving. Then it draws implications regarding the design of a techno mathematics curriculum for algebra at the collegiate level that establishes problem type schema. This design is supported by a philosophical basis of the role of technology in the acquisition of mathematical knowledge. The design is not instrument specific, since it is intended to be adaptable to whatever technology is available to both teachers and students be it in progressive countries or in the third world countries.

Recent research studies on mathematics education have placed its focus on the learners and their processes of learning. They have posited theories on how learners build tools that enable them to deal with problem situations in mathematics. Blais reveals that the philosophical and theoretical view of knowledge and learning embodied in constructivism offers hope that educational processes will be discovered that enable students to acquire deep understanding rather than superficial skills. (Blais, 1988, p.631)

As learners experience their power to construct their own knowledge, they achieve the satisfaction that mathematical expertise brings. They acquire the ability to engage in critical and analytical context of reflective thinking. They develop systematic and accurate thought in any mathematical process. O’Daffer and Thorquist (1993) define critical thinking as “a process of effectively using skills to help one make, evaluate and apply decisions about what to believe or do”(p.40). They cited the observations of Facett(1938) on a student using critical thinking as one who

1. Selects the significant words and phrases in any statement that is important and asks that they be carefully defined;

2. Requires evidence supporting conclusions she is pressed to accept;

3. Analyzes that evidence and distinguishes fact from assumption;

4. Recognizes stated and unstated assumptions essential to the conclusion;

5. Evaluates these assumptions, accepting some and rejecting others;

6. Evaluates the argument, accepting or rejecting the conclusion;

7. Constantly reexamines the assumptions which are behind her beliefs and actions.

Critical thinking abilities can only be developed in a setting which the learner has ample knowledge and experience. Thus, fostering critical thinking in a certain domain entails developing deep and meaningful learning within the domain. Learners can acquire critical thinking strategies by using what cognitive and developmental psychologists call a cognitive schema. Smith, Knudsvig and Walter (1998, p.50) describe a cognitive schema to be “a scheme, method, process by which (one) can see, organize and structure information” for better comprehension and recall. Through the schema learners interpret, analyze, organize and make sense of every information given in a problem situation through a constructive process called reflective abstraction.

Through reflective abstraction, critical thinkers are able to assimilate information into their mathematical network and build from their prior knowledge. They can accommodate new ideas including those that conflict with what they know or believe and negotiate these ideas. They are willing to adjust their belief systems after reexamining information. They are also able to generate new ideas based on novel ideas that are available to them. They are expert problem solvers who can handle abstract problem information and make sense of different problem situations.

On the other hand, novice problem solvers are not able to handle abstract mathematical concepts. They have difficulty recognizing underlying abstract structures and often need to make detailed comparisons between current and earlier problems before they can recognize the abstract information in the solution of the current problem ( Reed ,1987; Reed, Dempster, Ettinger, 1985; Anderson, 1984; Ross, 1987, as cited by Bernardo, 1994). They usually resort to algorithmic activity and not to the perception of essence. Blais (1988) observed that “they resist learning anything that is not part of the algorithms they depend on for success”(p.627). They tend to be very shallow in dealing with problem situations because of the lack of depth in their experiences while engaging in mathematical activities.

All problem solvers, whether experts or novices, develop a cognitive schema which cognitive scientists call problem-type schemata when confronted with a mathematical problem. According to Bernardo (1994), “[k]nowledge about the problem categories include information about the relevant underlying principles, concepts, relations, procedures, rules, operations and so on”(p.379). Further, he adds, “problem-type schemata are acquired through some inductive or generalization process involving comparisons among similar or analogous problems of one type”(p.379). Learners represent, categorize and associate problems to be able to determine the appropriate solution. The expert’s schematic processing leads to an accurate analysis of the problem which the novice hardly achieves. Bernardo (1994) claims that “the novices’ schemata (expectedly) include[s] mainly typical surface-level information associated with a problem type, whereas experts’ schemata include[s] mainly statements of abstract principles that [are] relevant to the problem type”(p.380). One example of the difference in the processing of experts and novices given by Blais (1988) is on their reading process of a mathematical material. Blais (1988) observes that, [w]hen novices read, the process almost always appears to be directed toward the acquisition of specific information that will be needed for algorithmic activity, (whereas) the reading process used by experts is directed toward the perception of essence. (p.624)

Experts seem to readily categorize the mathematical information in the material being read, thus facilitating the processing of information that lead to the correct solution. They are able to attain some sort of a visual form of say an algebraic expression and are able to communicate this before they perform the algorithmic activity. Besides, they can determine errors and attain a deep understanding of the underlying structure of the mathematical concept. Experts rely not only on concepts and procedures when confronted with a mathematical problem. They also have access to metacognition which is the knowledge used by experts in “planning, monitoring, controlling, selecting and evaluating cognitive activities” (Wong, 1989, Herrington, 1990, English, 1992 as cited by English-Halford, 1992; Bernardo, 1997). With this higher order thinking skill, problem solvers are assured of the success of every mathematical strategy they employ.

It is therefore the goal of education to help novices gain expertise in mathematical activities such as problem solving. In the next section, we deal with a few different views of studies conducted on didactics of problem solving.

Smith, Knudsvig and Walter (1998) advocate a cognitive schema which learners can use to acquire critical thinking strategies. They call it the TCDR for TOPIC-CLASS-DESCRIPTION-RELEVANCE. Thus, when given a learning material, students should ask the following questions:

· What TOPIC I must understand?

· What overall CLASS does this topic belong?

· What is the DESCRIPTION of the topic?

· What is the RELEVANCE of the topic?

These questions help learners interpret, analyze, organize and make sense of the information that are given in the material for better processing of learning. Once this becomes the framework of the learners, they gain strength and clarity of thinking. Several schemes have been offered by mathematics educators for solving word problems. The most versatile and widely used scheme for problem solving is the one formulated by George Polya (1957). These include working simpler problems, restating a problem, decomposing or recombining a problem, drawing figures, making charts or organized lists, exploring related problems, using logical deduction, using successive approximations, using guess-and-check methods, and working backwards. 

   

                      

Polya (1957) also developed a framework for problem solving in terms of such general phases as “understanding the problem, divising a plan, carrying out the plan and looking back” (cited by Barb and Quinn, 1997, p. 537). If carried out effectively, then the problem solver becomes successful in handling a problem situation. But the process involved in traversing these steps is quite complex. The learner has to use her prior knowledge, apply acquired mathematical skills, understand the context of the problem situation, and choose the appropriate strategy in solving the problem. This requires formal abstraction, a higher order thinking skill that is available to experts alone. What, then, can be done to help novices gain intellectual power?

By their success in working with simpler problems, novices gain confidence and are motivated to work with more difficult ones. Their analogical thinking can be best harnessed by using very concrete prior experiences. They are able to build their mathematical ideas from simple tasks and are able to acquire mathematical skills. Bernardo (1997) emphasizes the importance of the use of context problems that are familiar to the students which “provides students with a concrete (possibly, real) grounding on the problem, and which allows students to more easily draw from their existing knowledge about similar situations”(p. 11). Hopefully, students become more involved in the difficult task of making learning meaningful.

Mathematics educators recommend the use of mental models to guide learning. These mental models (aids) come in the form of diagrams or drawings used to represent the structure of the concept. The development of strategies and mental modeling fall under the theory of analogies. The effectivity of the analogy lies in a learner’s ability to recognize the “correspondence between the structure of the aid and the structure of the concept to be understood” (English-Halford, 1992, p. 121). In this case, the learner is able to map the essence of the model into the essence of the concept, and match or transfer specific conceptual aspects of one domain into another. This cognitive process promotes reflective abstraction. It is unfortunate, though, that certain popular pedagogical practices are counterproductive.

In the process of streamlining the problem solving task, teachers are sometimes tempted to use artificial and fabricated ways of building skills which Blais (1988) refers to as remedial processing. One good example is the prescription of finding key words in a problem which may work for experts, but not necessarily for novices. Some novices use these key words to decide on the algorithm to apply, with complete disregard of the essence of the problem. Key words prompt novices to add when they see the word increase, or subtract when they see the word decrease in a problem. Worse, some apply an arithmetic operation on any two numbers that they see depending upon the key words that they find in the problem. In fact, even their use of formal symbolic expressions in the solutions of the problems may not even communicate the essence of the given problem.

Blais (1988) laments that “[c]onventional instruction permits, allows, and sometimes blatantly encourages algorithmic activity that is separate and isolated from the perception of essence”(p. 627). This may be due to the focus of instruction on the product and not the process of the mathematical activity. In  fact, explanations sometimes send the wrong signal that problem solving processes are neat, well organized and easy as the teacher’s presentations on the board. Consequently, novices are tempted to resort to rote memorization of the algorithms, rules and formulas presented by the teacher. They do not realize that proficiency in problem solving is best achieved in recognizing the essence of a given problem and the application of the proper problem solving heuristics. Understanding the structural relations in a mathematical problem ushers the learners to reflective abstraction and gives them a sense of direction and feeling of certainty.

Barb and Quinn (1997) advocate the use of multiple methods of problem solving including such intuitively based methods as the guess-and-check method approximation. Problem solvers can use arithmetic computation with figures and charts and logical reasoning, and not necessarily algebraic equations in finding solutions. They believe that this strategy is more meaningful to a learner who is beginning to use some form of reflective abstraction, than rote application of algorithms usually found in textbooks. Teachers who usually look for algebraic solutions should be convinced of the value of developing the students’ problem-solving skills and refining their strategies using intuition and logic. It should be noted that the ultimate goal of this instructional method is to help learners build a good knowledge base in solving word problems so they can achieve reflective abstraction in the process. This belief was expressed by Owen and Sweller (1989) when they challenged the emphasis placed on problem solving and heuristics in the 1980s and pointed out that “superior problem solving performance does not derive from superior heuristics but from domain specific skills” (cited by Puut and Isaacs, 1992, p.215). They claim that the use of general cognitive strategies such as the means-end strategy impose heavy cognitive load and hamper schema acquisition and rule automation. It is because “a means - end tactic involves comparing the initial conditions of a task against the goal set for that task, then searching for a tactic that will transform either the goal or the initial conditions to be a bit more like one another” (Wine & Stockley, 1998, p. 124). This becomes very difficult especially when solving multistep problems. The solver has to analyze and break down the problem to subgoals, successfully transform each initial condition and subgoal into the desired condition, repeat the tactic until the final goal of the problem is achieved. The learner has to see the overall structure of information, concepts, operations, rules, and all other elements  that make up the whole schema of the problem. It is preferred that problems be freed of a single goal.

When the problem becomes goal-free, solvers are able to work forward from givens of the problem that they are able to generate. According to Wine & Stockley, “each iteration is a self-contained step that uses whichever problem-solving technique is easiest for the student, [in which case] the drain on working memory’s resources is minimized”(1998, p.125). In fact, Sweller (1989) claims that “research shows that freeing problems of singular goals can help students acquire schemas for solving problems “(cited by Wine & Stockley, p. 125). The development of domain-specific skills of learners may facilitate the development of schemas that underlie genuine understanding and meaningful learning.

Another issue that is worth considering is the question of when students should engage in word problems. Word problems are usually treated as application problems since they are given after certain mathematical concepts are introduced, with the aim of using the concepts in solving the problems. On the other hand, word problems may be taught in context, i.e. they may be used to teach a mathematical idea or process. According to Laughbaum (1999) “[t]eaching in context also uses problems or situations, but they are used at the beginning of a math topic for the purpose of helping students understand the mathematics to be taught, or to create a motivating experience of the mathematics to follow” (p.1). Certain groups looked into the effects of application problems to the development of the skills of the learners. One such group called the Cognition and Technology Group of Vanderbilt (CTGV) identified the shortcomings of the application problems and came up with efficient ways of teaching word problems in context. The CTGV has these to say about application problems:

1. Instead of bringing real world standards to the work, students seem to treat word problems mechanically and often fail to think about constraints imposed by real-world experiences.

2. Single correct answers to application problems lead to misconceptions about the nature of problem solving and inadvertently teaches students for a single answer rather than seek multiple answers.

3. The goal of one’s search for a solution is to retrieve previously presented information rather than rely on one’s own intuition. This may limit the development of people’s abilities to think for themselves.

4. They explicitly define the problems to be solved rather than help students to learn to generate and pose their own problems. Mathematical thinkers tend to generate their own problems.

5. The use of application problems lead to inert knowledge. Inert knowledge is that which is accessed only in a restricted set of contexts even though it is applicable to a wide variety of domain. (1997, p. 40)

These application problems are traditionally presented using general problem solving strategies which Polya prescribed or the means-end strategy. While some educators and researchers express the above mentioned concerns, many mathematics educators still adhere to the conventional practices of teaching problem solving. Lawson (1990), in defense of conventional methods, explained that when done properly, “general problem solving strategies play an important role in learning and transfer” (cited by English-Halford, 1992, p. 120). He described the three different types of general problem-solving strategies to include:

Task orientation strategies (which) influence the dispositional state of the student and include the broad affective, attitudinal, and attributional expectations held by the student about a particular task. Executive strategies are concerned with the planning and monitoring of cognitive activity, while domain-specific strategies include heuristics such as means-ends analysis and other procedures developed by the problem solver for organizing and transforming knowledge (e.g., constructing a table or drawing a diagram). (p. 120).

Lawson insisted that these strategies “have a general sphere of influence on cognitive activity during problem solving and should be seen as distinct from strategies specific to a particular task” (p. 404, 1990, cited by English-Halford, 1992, p.120).

Bernardo (1997) recommends the use of variable problem contexts to promote abstraction. He claims that “[b]y presenting concepts in variable problem contexts, students will come to appreciate the meaning and use of a particular concept or procedure in a variety of contexts”(p.12). Problem solvers cannot possibly recognize problem structure of single problems, thus the need for use of a wide range of diverse problems to facilitate the abstraction of specific concepts and transfer of knowledge to various problem contexts. He believes that a “deeper engagement of the problem information should lead to better conceptual understanding of the problem, and hopefully, result to higher level of abstract thinking about the problems”(p. 13). He proposes teaching strategies that promote analogical transfer. It should be noted that “many theorists argue that specific experiences are represented in memory as cases that are indexed and searched so that they can be applied analogically to new problems that occur”(Kolodner, 1991, Riesbeck and Schank, 1989; Schank, 1990 cited by CTGV, 1997, p.37). It is therefore the task of mathematics educators to determine ways of facilitating analogical transfer among learners.

One such instructional strategy that promotes analogical transfer involves presenting students with a context problem and then asking them to make their own problem using a different context. The effectiveness of this strategy according to Bernardo seems to be due to the deeper level of understanding of the problem structure achieved by the problem solver…[as she] explores the problem structure while attempting to create an analog, …[and] as a result of correctly mapping the problem structural information to create a true analog of the original problem. (Bernardo, 1998, p. 7)

Through this problem posing strategy the learners are able to recognize the essence of a problem and construct similar problems with the same essence. Mathematical problem posing, according to Silver (1994, cited by Ban-Har and Kaur, 1999) “is the generation of new problems or the re-formulating of existing ones ”(p. 77). It is recognized as “a valuable process that is motivating, challenging and allows students to exercise their creativity and independent learning skills” (Southwell, 1999; Silver, 1994, Kilpatrick, 1987 as cited by Ban-Har and Kaur, 1999). There are variety of ways to pose problems as a mathematical activity. These include writing questions based on given set of facts, on a given calculation, or on certain information. The benefits of the activity are the same whichever form is used. While results of recent studies give no clear correlation between quality of problem posing responses and problem solving ability (Ban-Har and Kaur, 1999), there are indications that, when performed in the context of analogical problem construction, analogical transfer is facilitated (Bernardo, 1998, p. 7).

There are other ways of facilitating recognition of problem structures, one of which is the use of text editing skills. In this activity, problem solvers are asked to identify missing information from problems or point out information that are irrelevant to the problems. Low and Over (1989) showed the significantly high correlation between students’ ability to edit the text of algebraic story problems and their ability to solve these problems; as well as between students’ ability to edit the text and categorize problems as being similar or different from each other (cited by Putt and Isaacs, 1992, p. 215). This activity enhances the problem solvers’ awareness of their own thinking processes. Such awareness helps learners identify their points of strengths and weaknesses and regulate their own ways of knowing. Garofalo and Lester (1985) claimed that “most problem solvers do not develop the appropriate metacognitive knowledge that should accompany the execution of computational procedures for doing  problems”(cited by Bernardo, 1997, p. 8). Wong (1989) and Herrington(1990) showed otherwise in their studies (cited by English-Halford, 1992). According to Wong (1989), “most students indicated that they were conscious of metacognitive processes and used strategies for monitoring and regulating the processes necessary for problem solving” (cited by English-Halford, 1992, p.118). Herrington (1990) also observed that “upper primary school children had well formed views on the process of learning mathematics and were able to confidently express them”(cited by English-Halford, 1992, p.119). Inspite of these varying opinions, Wong and Bernardo both agree about the need to use guided instruction in the use of metacognitive strategies for problem solving especially among lower ability students. Bernardo (1997) echoed Schoenfeld’s suggestion (1987, cited by Bernardo, 1997) that teachers model the metacognitive processes in problem solving when they present solutions to their students. A teacher thinks aloud and exhibits the process of planning, organizing, analyzing and carrying out the solution. The teacher articulates questions, makes mistakes, traces and corrects mistakes, deals with incorrect approach, backtracks, evaluates her progress, and struggles to arrive at the correct solution. This teaching strategy demonstrates the complexity of the process involved in solving problems and the reality that there are many possible ways of arriving at the correct answer.

In the light of all the issues and conflicts on various aspects of problem solving, particularly on developing cognitive strategies among students, and with the assumption that teachers hold wholesome beliefs and attitudes towards mathematics teaching, this paper attempts to offer suggestions on effective ways of fostering critical and analytical thinking through problem solving at different school levels.

At this point, we all agree that an expert problem solver is a critical and analytical thinker. When a learner gains expertise, she has acquired all the qualities of strong and smart thinking. She becomes insightful, and logical. The expert is also a constructive learner. She participates actively in the learning process and is able to build from her prior knowledge while assimilating and accommodating new knowledge. She appreciates the variety of ways of solving mathematical problems and recognizes a good solution. She is not afraid to use intuition and logic in her solutions. She makes good models of the problems and recognizes the essence and structure of a given problem. She employs a cognitive schema that helps her organize and plan her strategies. Her metacognitive skills help her monitor and evaluate her progress.

Expertise can be attained at an early age. Blais (1988) cites indicators of a schooler’s expertise once a teacher expresses doubt in her work. According to Blais, [I]f the child does not erase, if she or he refuses to accept the hint from an outside authority and tries to ponder whether the answer is correct, that student is an expert. Being willing and able to think and act independently, she or he will decide what is sensible and reasonable based on informal concepts already acquired (Mills, 1859). A child accustomed to accepting rules and procedures on faith has subordinated his or her own reasoning to outside authority and would have

yielded to it once again; the child would have erased. (Blais, 1988, p. 626) 

This suggests that teachers should allow their students to experience the joy of working independently by simply guiding and facilitating their learning and by not doing all the thinking and solving for them.

Writing problem-centred activities

‘Knowing mathematics is doing mathematics. We need to create situations where students can be active, creative, and responsive to the physical world. I believe that to learn mathematics, students must construct it for themselves. They can only do that by exploring, justifying, representing, discussing, using, describing, investigating, predicting, in short by being active in the world.’

There are several stages in writing problem-centred activities that provide opportunities for learners to develop, practise and apply their functional mathematics skills. Identifying a context or topic that is realistic and purposeful is the first and critical stage in writing problem-centred activities.

A subject teacher, working alone, may be able to identify contexts where functional mathematics opportunities occur naturally and can be used to contribute to the main subject learning. However, it can be very effective for a subject teacher and a functional mathematics expert to work in partnership to identify contexts and develop authoritative problems. This combination of knowledge and expertise will produce activities that achieve both aims: subject focus and mathematics process.

The following activity was devised by an electrical engineering supervisor and a functional mathematics teacher working together. While soundly rooted in the vocational context, it has been designed to ensure that learners have the opportunity to practise and apply their functional mathematics skills.

Resistance and temperature

Scenario

Your supervisor has given you the job of rewiring a house that has been heavily insulated in the loft. The resistance of wire increases with temperature. In the height of summer, the temperature in the loft can reach 50°C.

Task

Investigate whether the 2.5 mm² annealed copper wire usually used for this purpose will be suitable for this particular job. It is unlikely that many functional mathematics teachers will have the knowledge of the engineering context to be able to develop this activity on their own. The expertise of the engineering supervisor is critical to ensuring that the context is authentic but they will need guidance from the functional mathematics teacher if they are to get the level right.

Partnership teaching, where the subject specialist and the functional mathematics teacher work together to plan, prepare and deliver the lessons, can also be highly effective. This collaborative working enables the functional mathematics teacher to identify the mathematical skills that underpin the subject content and devise relevant activities that will provide opportunities for learners to practise and apply their functional mathematics. This, in turn, will help learners to achieve their core learning aims.

In other provision, the activity can be led by the learners’ expressed interests. The following activity has been devised to encourage a learner with a keen interest in competitive weightlifting to develop and apply mathematical skills.

Weightlifting competition

Scenario

You are taking part in a weightlifting competition in 12 weeks’ time and have high hopes of winning a trophy. You know that you will need to prepare a detailed programme of training for the coming weeks.

Task

Using the template provided, plan a programme of training for the 12 weeks that will ensure that you are at peak fitness for the competition. The activity could be adapted for a learner who is keen on another sport.

Another way to challenge learners to apply their functional mathematics skills is to suggest that they develop an activity for another learner. This will enable them to identify and demonstrate their knowledge and understanding of the functional mathematics skills as well as using a range of other skills.

Dimensions and deviations

Scenario

Many thousands of bricks are used every day in the construction industry. Bricks should be made to British Standard 3921:1985*, which outlines the coordinating size and work size of bricks and the dimensional deviation allowed.

Task

In your workplace, test a sample of bricks to determine if they meet the British Standard.

*You would need to ensure that learners are aware of and have access to the relevant British Standard. This type of activity could be usefully introduced after learners have covered the knowledge about brick sizes and British Standards in their subject learning.

Everyday topics that are of interest to learners can be used as a basis for activities, for example deciding on the best buy from a range of options; fundraising to support a local environmental issue; finding the cost of a holiday for four.

New home for newts

Scenario

An article in your local paper highlights the threat to the population of newts in the park near your home due to the rubbish being dumped by visitors. You decide to enlist the help of friends to raise £100 to clean up the pond and make it environmentally safe for the newts.

Task

Consider the different ways you could raise the money and decide on the best option to put to your friends.

• Learners themselves can provide ideas for functional mathematics. Many learners will have an interest that can be harnessed to provide a scenario for an activity, for example sports, hobbies, leisure pursuits, cars. The following activity was devised for a learner who had a keen interest in Formula One motor racing.

Formula One

Scenario

Formula One is a very expensive sport. It is important that a team does well in the Constructors’ Championship if their sponsors are to continue to provide the money to run the team.

Task

Investigate how the different teams in Formula One have performed over the last ten years. Present your findings to a group of your fellow students. Decide which team you would advise a sponsor to support next year and explain your reasons.

Many purposeful activities will enable learners to use a range of functional skills, not just in mathematics. In many of the activities in this section there are opportunities for learners to develop and practise ICT skills, for example searching the internet, producing a graph to present data, and English skills, such as writing a questionnaire, producing a report, engaging in discussion.In the real world, functional skills are not used in isolation. Making the links with other functional skills will increase learners’ opportunities to use and apply the skills in a range of contexts.

Having decided on the context or topic, you will need to consider:

• the purpose of the activity

• the scenario

• what learners will be expected to do

• what results learners should produce

• how the functional mathematics process skills of representing,

analysing and interpreting can be used to complete the activity. Presentation and the use of language are important. Activities should be concise and should be written in language that learners will understand. Structure, style and readability are all important.

If you are developing a number of activities, it will help learners in the early stages if they all contain the same elements and follow the same format; learners will know what to expect. However, as learners become more

confident in their mathematical skills, it may be appropriate to devise activities that are less structured, allowing learners to use a greater degree of autonomy to determine the processes and information required. This is particularly relevant for higher-level learners, but it is worth noting that real world problems tend not to come in a structured format, so learners at every level will benefit from being provided with activities that vary in the degree of structure.

A structured activity should include:

• the title – short and informative, but exciting the interest of the learner

• an overview – this may be through a scenario, but should set the scene for the activity and outline the context

• the task or tasks – what the learner has to do

• the outcome – what the end result should be; what the learner should produce

• resources – any materials required to complete the activity, or guidance as to where they can be found, as appropriate to the level of the learner

• the timescale for delivery of the completed task.

It may also be useful to show the links to main subjects, if appropriate.

Key points for good style include the following.

• Address the learner directly in your writing – use ‘you’.

• Avoid the passive wherever possible.

• Avoid ambiguity. It is easy to make assumptions – you know what is involved but learners may not.

• Keep paragraphs short and to the point.

• Use simple direct words rather than complex or formal language.

• Use short sentences.

• Use words that learners are likely to know and understand. Define any new technical terms at the first time of use.

• Use bullets or numbered lists as appropriate.

An activity should be clear, easy to read and attractive to learners. Here are some Do’s and Don’ts to keep in mind.

• Don’t try to fit too much on a page.

• Do avoid clutter; use lots of white space.

• Do use a typeface that is easy to read.

• Don’t use lots of different typefaces.

• Do use headings where appropriate, but don’t use unnecessary capital letters.

• Don’t use images that are not relevant to the activity.

A task or activity should be considered in terms of its level of demand in relation to its complexity, familiarity, technical demand and the independence required of the learner. An activity can be differentiated for different levels and, where appropriate, to meet the needs of individual learners. For example, activities could provide

more or less learner support. For lower-level learners this might mean providing additional guidance in the form of resources, more teacher input, or breaking down the task into smaller chunks. For higher-level learners it could mean more demanding outcomes, greater complexity in the tasks required, and multi-stage interrelated tasks.

This stage also provides an opportunity to identify links with other functional skills. The following activity requires functional mathematics, but learners will be able to approach the activity with more confidence if they are able to apply functional English and ICT skills as well.

Travelling on the job

Scenario

You have just got a new job in the London area as a computer technician. It will mean regular travel by car to offices in Central London (SW1), Slough, Epsom, Bexley, Chigwell and Watford.

Task

Your employer has promised to help with the removal costs to relocate from Newcastle if you can provide an estimate. Where would be the best place to live, taking into account the travel costs and time taken?

The activity can be adapted to make it more relevant for learners by using local places and contexts.

Learners could use their functional ICT skills to search for and determine the mileages between the offices using a route planner. They could also use their functional ICT and functional English skills to present the final estimate to the employer. With all problem-centred activities, it is important to give effective formative feedback, as this will enable learners to confirm their skills and identify areas for further development. It is worth identifying opportunities for formative assessment at this stage of the writing process.

Learners’ feedback on an activity will help to tell you whether it was effective, although you will need to be clear about the type of feedback you want. You could ask learners for their opinion of an activity informally through group discussion, or formally through a short questionnaire.

With support and encouragement, learners are usually keen to express their views. Key areas you might want to ask them about include the following.

• Did you find the activity easy to understand?

• Did you have or could you find all the source material you needed to complete the activity?

• What skills did you use to complete the activity?

• What did you like about the activity?

• Was there anything you didn’t like or found difficult?

• What did you learn from the activity?

This feedback will enable you to revise or adapt the activity to meet different learners’ needs and interests. It may also generate further ideas for activities and engage learners in the development process. In fact, you could add a final question asking learners for suggestions that could be used for writing future activities.

• Be open to finding sources of ideas and activities in the most unlikely places.

• Listen to learners – they may provide ideas or topics of interest that you can develop.

• Be aware of links with other functional skills and other subjects, courses and programmes.

• Working in partnership with a colleague can enhance the process of writing activities and increase the stock of ideas.

Learners who are functional in their mathematics can transfer their mathematical skills to a wide range of contexts. They can select the appropriate techniques and carry out calculations to solve different problems. They can use mathematics in their everyday lives, including in their work at school or college, in their jobs, in making shopping decisions, or in managing their personal finances.