Specific Instructional Techniques for Specific Instructional Goals

Once a classroom teacher is clear about the instructional goal within a unit, he or she should identify specific instructional techniques for specific types of knowledge. Even though this meta-analysis cannot yet be considered complete, it suggests that the instructional techniques identified are most effective for the various possible instructional goals.

Knowledge Goals

If the instructional goal is to enhance students’ understanding of vocabulary terms and phrases:

1. Provide students with a brief description or informal definition of each word or phrase.

2. Have students describe the words or phrases in their own words and represent their personal descriptions using some form of nonlinguistic modality (e.g., pictures, semantic maps, charts).

3. Occasionally have students review the terms and phrases making refinements in their representation. 

If the instructional goal is to enhance students’ understanding of details:

a) Present the details in some form of story or elaborated description.

b) Have students represent their understanding of the details in linguistic (e.g., notes, outlines) and nonlinguistic formats (e.g., pictures, semantic maps, charts, etc.). 

If the instructional goal is to enhance students’ understanding of organizing ideas (e.g., concepts, generalizations, principles):

a) Demonstrate the organizing ideas to students in concrete terms.

b) Have students apply the concept, generalization, or principle to new situations. 

If the instructional goal is to enhance students’ ability to perform subject-specific algorithms:

a)Present the various steps in the algorithm.

b) Have students practice the algorithm paying particular attention to how it might be improved.

If the instructional goal is to enhance students’ ability to perform subject-specific tactics or processes:

a) Present students with general rules or heuristics as opposed to specific steps.

b) Have students practice the tactic or process paying particular attention to how it might be improved.

If the goal is to enhance students’ ability to perform psychomotor skills:

a) Present students with a model of the psychomotor skill.

b) Have students practice the skill paying particular attention to how it might be improved.

Cognitive Goals

If the goal is to enhance students’ ability to store and retrieve knowledge:

a) Provide students with strategies that use the representation of knowledge in non linguistic forms (e.g., mental images).

 If the goal is to enhance students’ ability to identify similarities and differences, to analyze the reasonableness of new knowledge, to generate inferences about new knowledge, or to apply organizing ideas:

a) Provide students with a set of heuristics, as opposed to steps regarding the processes involved.

b) Have students practice the heuristics, paying particular attention to how they might be improved.

If the goal is to enhance students’ ability to represent knowledge in a variety of forms:

a) Provide students with strategies for representing knowledge linguistically.

b)Provide students with strategies for representing knowledge non linguistically.

If the goal is to enhance students’ ability to comprehend information presented orally (i.e., listening):

a) Present students with a set of heuristics, as opposed to steps for the overall process of listening.

b) Have students practice the heuristics, paying particular attention to how they might be improved.

If the goal is to enhance students’ ability to comprehend information presented in written form:

a) Provide students with information and strategies designed to enhance their ability to decode print. Have them practice the strategies, paying particular attention to how they might be improved.

b) Provide students with a set of heuristics for the overall process of reading. Have students practice the heuristics, paying particular attention to how they might be improved. 

c) Provide students with strategies for activating what they know about a topic prior to reading.

d) Provide students with strategies for summarizing information they have read.

e) Provide students with information about the various text formats they will encounter.

f) Provide students with strategies for representing what they have read in non linguistic form and as mental images.

If the goal is to enhance students’ ability to present information in oral form (i.e., speak):

a) Present students with information about the various conventions used in different situations.

b) Provide students with heuristics for the overall process of speaking in various situations, and have them practice the heuristics, paying particular attention to how they might be improved.

c) Provide students with strategies for analyzing a topic in depth prior to speaking about it.

If the goal is to enhance students’ ability to present information in written form:

a) Provide students with heuristics for the overall process of writing, and have students practice these heuristics, paying particular attention to how they might be improved.

b) Present students with strategies for encoding thought into print.

d) Present students with strategies for analyzing a topic in depth prior to writing about it.

e) Provide students with information about the various discourse formats in which they will be

expected to communicate.

If the goal is to enhance students’ ability to make decisions, solve problems, or perform investigations:

a) Provide students with heuristics for the overall processes of decision-making, problem solving, and investigation, and have them practice the heuristics, paying particular attention to how they might be improved.

b) Provide students with strategies for using what they know about the topics that are the focus of problems, decisions, and investigations.

If the goal is to enhance students’ ability to engage in experimental inquiry:

a) Provide students with heuristics for the overall process of experimental inquiry, and have them practice the heuristics, paying particular attention to how they might be improved.

b) Provide students with strategies for generating and testing hypotheses.

c) Have students apply the experimental inquiry process to a variety of situations.

Metacognitive Goals

If the goal is to enhance students’ ability to set explicit goals, identify strategies for accomplishing goals, or monitor progress toward goals:

1. Have students verbalize their thinking as they engage in these functions, and analyze the effectiveness of their thought processes.

2. Present students with information about the nature and importance of using the metacognitive system.

If the goal is to enhance students’ ability to monitor their use of the various dispositions:

1. Provide students with explicit information about the nature and function of the various dispositions.

Self Goals

If the goal is to enhance students’ understanding of and control of their beliefs about self attributes, self and others, the nature of the world, efficacy, or purpose:

1. Have students verbalize their thinking relative to these areas.

2. Have students make linkages between specific beliefs and specific behaviors in their lives.

3. Have students identify those behaviors they wish to change.

4. Provide students with strategies for altering their thinking relative to the behaviors they would like to change.

Knowledge Utilization Processes

There are four knowledge utilization processes: (1) decision-making, (2) problem solving, (3) experimental inquiry, and (4) investigation.

Decision-making: The process of decision-making is utilized when an individual must select between two or more alternatives (Baron, 1982, 1985; Halpern, 1984). The execution of the decision-making process requires an individual to retrieve from permanent memory his prior knowledge about the topic. For example, if the individual is going to make a decision regarding where to go on a Sunday afternoon pleasure drive, he will retrieve what he knows about local destinations. He will also retrieve what he knows about the various steps and heuristics involved in the overall process of decision-making. Steps and heuristics commonly associated with the overall process of decision-making include:

1. identification of alternatives

2.assigning of values to alternatives

3.determination of probability of success

4.determination of alternatives with highest value and highest probability of success

The execution of each of these steps requires the individual to analyze the data in working memory using the basic information processing functions. For example, to assign value to alternatives, an individual must use the matching function to determine the similarities and differences between the characteristics of the alternatives that have been identified and the characteristics of a "successful Sunday drive" as defined by the individual.

In summary, decision-making involves the following aspects:

1. Activation, via retrieval of knowledge about the topic & the alternatives under consideration and personal values related to those alternatives

2. Activation, via the retrieval function, of knowledge about the overall process involved in decision-making

3. Analysis of data in working memory via the information processing functions

Problem-solving: The process of problem-solving is utilized when an individual is attempting to accomplish a goal for which an obstacle exists (Rowe, 1985). As is the case with decision-making, problem-solving requires the activation of prior knowledge about the topic. For example, if an individual wishes to be at a specific location some miles from her home by a certain time and her car breaks down, she has a problem & she is attempting to accomplish a goal (i.e., to transport herself to a specific location) and an obstacle has arisen (i.e., her usual mode of transportation is not available). To address this problem effectively, the individual would have to retrieve from permanent memory her prior knowledge about different methods of transportation that are alternatives to taking her car (e.g., taking the bus, calling a friend) as well as options for fixing her car within the available time. In addition to knowledge about the topic, the individual would have to retrieve her knowledge about the overall process of problem-solving. Steps and heuristics commonly associated with problem-solving include:

1. identification of obstacle to goal

2. possible re-analysis of goal

3. identification of alternatives

4. evaluation of alternatives

5. selection and execution of alternatives

(Halpern, 1984; Rowe, 1985; Sternberg, 1987)

Again, the execution of each of these steps or heuristics requires the individual to analyze the data in working memory using the information processing functions. Of these functions, information specification and information screening are probably key factors. An individual uses the specifier to generate hypotheses about possible ways to overcome the obstacle. The information screen-er is used to evaluate the feasibility and likelihood of the options that have been generated. In summary, problem-solving involves the following components:

1. Activation via retrieval of knowledge about the topic

2. Activation via retrieval of knowledge about the overall process involved in problem-solving process

3. Analysis of data in working memory via the information processing function

Experimental inquiry: Experimental Inquiry involves generating and testing hypotheses for the purpose of understanding some physical or psychological phenomenon. To engage in experimental inquiry, an individual must activate knowledge of the topic. For example, if an individual has a question about how airplanes fly, she will activate her knowledge of concepts important to the phenomenon under investigation such as lift and drag. Additionally, she will retrieve from permanent memory knowledge of the steps and heuristics involved in the process of experimental inquiry. The steps and heuristics commonly associated with the experimental hypothesis include:

1. making predictions based on known or hypothesized principles

2. designing a way to test the predictions

3. evaluating the validity of the principles based on the outcome of the test

                                                                                                        (Halpern, 1984; Ross, 1988)

The execution of these steps requires the individual to analyze the data in working memory using the information processing function. Of these functions, information specification and information screening are probably key. For example, the information specification function would be used to generate predictions based on known principles about lift and drag. The information screening function would be used to judge the reasonableness of the results of the experiment given the individual’s initial understanding of the concepts of lift and drag.

In summary, the experimental inquiry process involves the following components:

1. Activation via retrieval of knowledge about the topic under investigation

2. Activation via retrieval of knowledge about the overall process involved in experimental inquiry

3. Analysis of data in working memory via the information processing function.

Investigation: It is the process of generating and testing hypotheses about past, present, or future events. It is similar to experimental inquiry in that hypotheses are generated and tested. It differs from experimental inquiry in that it utilizes different "rules of evidence." Specifically, the rules of evidence for investigation adhere to the criteria for sound argumentation like the establishment of warrants (Toulmin, 1958; Toulmin, Rieke and Janik, 1981), whereas the "rules of evidence" for experimental inquiry adhere to the criteria for statistical hypotheses testing. In short, the investigation process can be conceptualized in the same way as the experimental inquiry process:

1. Activation via retrieval of knowledge about the topic

2. Activation via retrieval of knowledge about the overall process involved in investigation

3. Analysis of data in working memory via the information processing function

From the discussion above, it should be clear that all the knowledge utilization processes have the same basic syntax. Specifically, they require individuals to retrieve knowledge of the topic from permanent memory as well as knowledge of the overall process involved. Additionally, they all make heavy use of the information processing functions to analyze the data in working memory throughout the execution of the overall process. The primary difference in the knowledge utilization processes is in the steps and heuristics that define the overall process and the information processing functions they most heavily employ.

The Modularity of the Human Mind

A basic assumption regarding the three modalities described in this chapter is that experiences can and frequently are encoded in memory using all three modalities. That is, experiences are stored or encoded as three dimensional "packets." This modularity assumption is quite consistent with current brain theory. Anderson (1995) refers to these modular encoding of experience as "records." The concept that the brain has a modular structure has received a great deal of attention over the last two decades. Gazzaniga (1985) describes the modularity of the brain in the following way:

By modularity, I mean that the brain is organized into relatively independent functioning units that work in parallel. The mind is not an indivisible whole operating in a single way to solve all problems . . . the vast and rich information impinging on our brain is broken into parts . . . (p. 4) In his book The Modular Brain (1994), Richard Restak details the historical development of the concept of brain modularity, noting that over time it replaced the theory that the brain has a strict hierarchic organization. Restak credits Johns Hopkins neuroscientist  Vernon Mountcastle as the primary architect of the modern modular theory. He cites Mountcastle as saying:

. . . Most people don’t think hierarchically anymore; they shy away from saying "This function resides here." Instead, we now believe the brain is arranged according to a distributed system composed of large numbers of modular elements linked together. That means the information flow through such a system may follow a number of different pathways, and the dominance of one path or another is a dynamic and changing property of the system. (Restak, 1994, p. 35) 

The discussion of modularity by Restak, Mountcastle, and Gazzaniga addresses the physical architecture of the brain. Here I refer to the psychological structure of the mind & the structure of stored experience. As discussed previously, a basic assumption of the theory presented in this book is that human experiences are stored in three dimensional modules. For example, if a person goes to a football game, the experience will be encoded as mental pictures, smells, tastes, sounds, and kinesthetic sensations & all forms of non linguistic representation. The individual might also store emotions associated with the game such as anger or joy & forms of the affective modality. Finally, the individual will encode the experience as deep-structure propositions describing what occurred & forms of the linguistic modality. Some psychologists believe that, over time, the linguistic modality becomes the dominant mode of processing. "The behaviors that these separate systems emit are monitored by the one system we come to use more and more, namely the verbal natural language system" (Gazzaniga and LeDoux, 1978, p. 150). The primacy of language was demonstrated in a study by Mandler and Ritchey (1977). Subjects were shown ten pictures, one right after another. Pictures contained fairly common scenes like a teacher at a blackboard with a student working at a desk. After viewing eight such pictures for ten seconds each, subjects were presented with a series of pictures and asked to identify which ones they had seen. The series contained the exact pictures they had studied as well as distraction pictures. Two types of distractors were used, token distractors and semantic changes. Token distractor pictures changed details of the target pictures like the pattern in the teacher’s dress. Pictures that contained semantic changes altered some element that was at a high level of importance in terms of the propositional network representing the picture. For example, a semantic change might change the teacher from a male to a female. There was no systematic difference in the amount of physical change in the pictures between token changes and semantic changes. Subjects recognized the original pictures 77 percent of the time, rejected the token distractors only 60 percent of the time, but rejected the semantic distractors 94 percent of the time. In short, subjects had encoded each picture as a linguistically-based set of abstract propositions with an accompanying visual representation. It was the propositional changes in the picture that were best recognized, not the changes in the nonlinguistic aspects of the information.

Conversely, arguments are also made that the emotional system is the primary representational modality. Specifically, a good case can be made for the assertion that the affective modality exerts the most influence over human thought and experience. This case is well articulated in LeDoux’s The Emotional Brain: The Mysterious Underpinnings of Emotional Life (1996). Among other things, as a result of his analysis of the research on emotions, LeDoux concludes that human beings 1) have little direct control over their emotional reactions, and 2) once emotions occur, they become powerful motivators of future behavior. Relative to humans’ lack of control over emotions LeDoux notes:

Anyone who has tried to fake an emotion, or who has been the recipient of a faked one, knows all too well the futility of the attempt. While conscious control over emotions is weak, emotions can flood consciousness. This is so because the wiring of the brain at this point in our evolutionary history is such that connections from the emotional systems to the cognitive systems are stronger than connections from the cognitive systems to the emotional systems. (p. 19)

Relative to the power of emotions once they occur, Le Doux explains:

They chart the course of moment-to-moment action as well as set the sails toward long-term achievements. But our emotions can also get us into trouble. When fear becomes anxiety, desire gives way to greed, or annoyance turns to anger, anger to hatred, friendship to envy, love to obsession, or pleasure to addiction, our emotions start working against us. Mental health is maintained by emotional hygiene, and mental problems, to a large extent, reflect a breakdown of emotional order. Emotions can have both useful and pathological consequences. (pp. 19-20) For LeDoux, then, emotions are primary motivators that often outstrip an individual’s system of values and beliefs relative to their influence on human behavior. This was demonstrated in a study by Nisbett and Wilson (1977) who found that people are often mistaken about internal causes of their actions and feelings. The researchers noted that individuals always provide reasons for their actions. However, when reasoned and plausible reasons are not available, people make up reasons and believe them. As described by LeDoux, this illustrates that the forces that drive human behavior cannot be attributed to the rational conclusions generated by our linguistic mind, but are functions of the inner workings of our emotional mind.

Learning Progressions - Curriculum, Instruction and Assessment Design

One of the major challenges to bringing mathematics, as well as most emerging science, into the classroom is their interdisciplinary nature.  However, students often have difficulty making connections between different scientific concepts and ideas. For instance, students often have difficulty applying knowledge from one part of the particulate model of matter to another (Renström, Andersson, & Marton,1990). In addition, students often use models of different levels to describe different concepts related to the structure and behavior of matter. (Harrison & Treagust, 2000).

The integration of knowledge is made more difficult by typical large-scale and classroom assessments ostensibly based on the standards. Such assessments commonly focus on targeted, isolated topics that do not require students to connect currently taught concepts with concepts from other science areas that were previously learned (NRC, 2005; Pellegrino, Chudowsky, & Glaser, 2001). Instead, these assessments encourage teachers to focus on isolated bodies of knowledge that ultimately results in compartmentalized application of science concepts. As a result, the traditional curriculum often compartmentalizes the various aspects of the study of matter (e.g. structure of matter, conservation of matter, chemical reactions, phase changes). The authors of documents such as Benchmarks for Science Literacy (AAAS, 1993) and the National Science Education Standards (NRC, 1996) suggested connections between key concepts among multiple disciplines in the sciences. However, these connections have not been borne out in most science curricula nor are they a part of typical assessment practices. Thus, in order to generate literacy in emerging sciences, school curricula must begin to emphasize not only the learning of individual topics, but also the connections between them and assessments must be developed to support such a curriculum.

This study describes work towards developing and validating the sequence and assumptions behind a learning progression. The processes of assessment that we use might ultimately be translated into both classroom and large-scale assessment strategies/materials. In addition, the work informs both the curricular structure and instruction by providing insight into how students connect ideas from other science disciplines with a core scientific concept. Thus, this approach might provide a method for identifying the connections that are required to obtain a deep conceptual understanding of an interdisciplinary field such as nano science. A learning progression describes what it means to move towards more expert understanding in an area and gauges students’ increased competence related to a core concept or a scientific practice (Smith et al., 2004). They consist of a sequence of successively more complex ways of thinking about an idea that might reasonably follow one another in the process of students developing understanding about that idea. However, as we address interdisciplinary subject matter, we can no longer consider learning progressions in a linear fashion. Rather, learning progressions may be viewed as strategic sequencing that promotes both branching out and forming connections between ideas related to a core scientific concept. We can better assess students’ conceptual understanding by designing items that assess the connections between

related science topics and ideas.

In order to provide a conceptual explanation of most nano scale phenomena, a deep and thorough understanding about the nature of matter is required. This includes the structure and properties of matter and how it behaves under a variety of conditions. To make progress on how students’ understanding about the nature of matter develops, we need to construct a learning progression.  However, this progression is only theoretical as it is does not represent the curriculum that is followed in the classroom nor do we have much empirical evidence to support this progression. Therefore, although the suggested progression is consistent with national standards, it requires collection of empirical evidence to verify a possible learning progression that can be applied to describe the learning of our target populations. This study is directed towards assessing and characterizing how and when students make the connections between the ideas as they progress towards a deep conceptual understanding. It affords practical guideline to develop test items for measuring students’ learning progression. As we study student learning of the nature of matter, we explore these questions:

1. How do students’ ideas about concepts regarding the nature of matter develop over time?

2. How can we assess the requisite connections between concepts that traditionally have been compartmentalized in instruction?

In this article I try to present how we can develop and validate a potential learning progression to use for the development of assessing student learning in nano science, specifically in the topic areas involving the structure, properties and behavior of matter. Our work is a case study that illustrates the design of assessments based on a learning progression through research and development cycles. Our principles include (a) elaborating standards and Benchmarks to create a hypothetical learning progression, (b) collecting various data to validate the learning progression, (c) revising the learning progression based on the collected data, and (d) developing items for assessing student’s placement on the learning progression. The difficulties that we have identified in students’ conceptual understanding of the nature of matter are not necessarily because the material is developmentally inappropriate. As more learning progressions are developed and validated, they can be used to guide teaching and curriculum development. If students were presented with an exemplary curriculum that helped to foster their understanding and facilitate the connections they must make between ideas to have a deep understanding of a science topic, the problems we observed might not occur.

The SOLO Taxonomy

The SOLO Taxonomy is based on the study of outcomes of academic teaching. SOLO is short for “Structure of the Observed Learning Outcome” and the taxonomy names and distinguishes five different levels according to the cognitive processes required to obtain them: “SOLO describes a hierarchy where each partial construction [level] becomes a foundation on which further learning is built” (Biggs, 2003, p. 41). SOLO can be used to define ILOs, forms of teaching that support them, and forms of assessment that evaluate to what extent the ILOs were achieved. It is developed aiming at research-based university teaching as the research activities behind it ultimately converge on real research (i.e. on the production of new knowledge) at its fifth and highest level. The five levels are as follows, in increasing order of structural complexity (Biggs & Collis, 1982, pp. 17-31; Biggs, 2003, pp. 34-53; Biggs & Tang, 2007, pp. 76-80):

SOLO 1: “The Pre-Structural Level”

Here the student does not have any kind of understanding but uses irrelevant information and/or misses the point altogether. Scattered pieces of information may have been acquired, but they are unorganized, unstructured, and essentially void of actual content or relation to a topic or problem.

SOLO 2: “The Uni-Structural Level”

The student can deal with one single aspect and make obvious connections. The student can use terminology, recite (remember things), perform simple instructions/algorithms, paraphrase, identify, name, count, etc.

SOLO 3: “The Multi-Structural Level”

At this level the student can deal with several aspects but these are considered independently and not in connection. Metaphorically speaking; the student sees the many trees, but not the forest. He is able to enumerate, describe, classify, combine, apply methods, structure, execute procedures, etc.

SOLO 4: “The Relational Level”

At level four, the student may understand relations between several aspects and how they might fit together to form a whole. The understanding forms a structure and now he does see how the many trees form a forest. A student may thus have the competence to compare, relate, analyze, apply theory, explain in terms of cause and effect, etc.

SOLO 5: “The Extended Abstract Level”

At this level, which is the highest, a student may generalize structure beyond what was given, may perceive structure from many different perspectives, and transfer ideas to new areas. He may have the competence to generalize, hypothesize, criticize, theorize, etc. We define competence progression as moving up the SOLO levels; i.e. SOLO-progression. Surface learning (which has similarities to instrumental understanding) implies that the student is confined to action at the lower SOLO levels (2-3); whereas deep learning (which has similarities to relational understanding) implies that the student can act at any SOLO level (2-5), including the higher levels (4-5). As we move up the SOLO hierarchy, we first see quantitative improvements as the student becomes able to deal with first a single aspect (from 1-2) and then more aspects (from 2-3). Later we see qualitative improvements (from 3-4) as the details integrate to form a structure; and (from 4- 5) as the structure is generalized and the student can deal with information that was not given. For these reasons, the levels 2 and 3 are sometimes referred to as quantitative levels; levels 4 and 5 as the qualitative.

Taxonomies for understanding - “ What is understanding”

The purpose of  teaching is inter alia that students should learn something; i.e., they should attain some level of understanding and skills. However, the term ‘understanding’ is used for many different things such as one student’s capacity to name main concepts involved in topic X and another student’s critical comparison of practical implications of theoretic models of topic X. These two uses of ‘understanding’ are different and embody both surface and deep understanding, respectively. According to Wittgenstein, words get their meaning from their use: “Nur in der Praxis einer Sprache kann ein Wort Bedeutung haben” (Wittgenstein, 1991, p. 344), but the usage of ‘understanding’ is ambiguous, hence its meaning is not clear. In Sausurre’s (1997, pp. 12-13) terminology, one could say that the term ‘understanding’ is a double entity constituted by one distinct succession of syllables (syllables), but multiple meanings (signification) linked to the syllables. Some clarity is therefore needed when one for instance explains the kind of understanding of topic X that is intended by the teacher and that will be tested at an examination. Skemp (1987, pp. 152-163) defined two types of understanding. ‘Instrumental understanding’ is “rules without reasons” for instance that you ‘understand’ that to divide a fraction by a fraction “you turn it upside down and multiply”. ‘Relational understanding’ occurs when one has built up a conceptual structure (schema) of topic X and therefore both know what to do and why when one solves a problem within that topic. However, Skemp’s distinction does not formulate a gradual development or levels of understanding.

 There are several taxonomies describing various levels of understanding. Gall (1970) presented an overview of eight of these of which Bloom’s is probably the most well known. These have been developed inter alia to classify questions “based on the type of cognitive process required to answer the question” (Gall, 1970, p. 708). Gall furthermore stated: “I have organised the categories to show similarities between the systems. It appears that Bloom’s Taxonomy best represents the commonalities that exist among the systems” (Gall, 1970, p. 710). Lewis (2007) gave a rather similar overview of five taxonomies. 

These taxonomies have been further developed but the basic ideas are still the same and particularly Bloom’s Taxonomy is still very widely used. However, they were not developed specifically with university teaching in mind and furthermore Bloom did not make his taxonomy with the purpose of formulating ILOs but to be able to select representative tasks for an examination (Biggs & Collis, 1982, p. 13). Below we therefore present a taxonomy for understanding “understanding” particularly aimed at assessing university students’ competencies. Following the discussion from above, this taxonomy distinguishes between “learn (to do)” and “learn about”. Lists of “learn about” are content statements using nouns listing the concepts and areas of knowledge that the students will encounter during the course. But this is not the same as what they “learn (to do)”. Curricula writers need to ask themselves what they want the students to get out of meeting these areas of knowledge; i.e., what do they want the students to learn to do? When assessing a student, we cannot actually measure the student’s knowledge “inside the brain”. What we can do, however, is to have a student do something, and then measure the product and/or the process. Therefore, it is important to focus on what the student does and on what the students are supposed to “learn (to do)”, i.e. what competencies the students are expected to have by the end of the course. As generally advocated in Outcomes-Based Education (OBE); in particular, in Constructive Alignment (Biggs, 2003), we therefore focused on having course descriptions with ILOs formulated using verbs stating what the students should be able to do by the end of the course. Having these things made explicit furthermore makes it easier to explain to the students what they are supposed to get out of a course.

Learning progressions

By its very nature, learning involves progression. To assist in its emergence, teachers need to understand the pathways along which students are expected to progress. These pathways or progressions ground both instruction and assessment. Yet, despite a plethora of standards and curricula, many teachers are unclear about how learning progresses in specific domains. This is an undesirable situation for teaching and learning, and one that particularly affects teachers’ ability to engage in formative assessment.

The purpose of formative assessment is to provide feedback to teachers and students during the course of learning about the gap between students’ current and desired performance so that action can be taken to close the gap. To do this effectively, teachers need to have in mind a continuum of how learning develops in any particular knowledge domain so that they are able to locate students’ current learning status and decide on pedagogical action to move students’ learning forward. Learning progressions that clearly articulate a progression of learning in a domain can provide the big picture of what is to be learned, support instructional planning, and act as a touchstone for formative assessment.

Learning progressions define the pathway along which students are expected to progress in a domain. They identify the enabling knowledge and skills students‘ need to reach the learning goal as well as provide a map of future learning opportunities. Heritage, Kim, Vendlinski, and Herman (2009) explain that learning progressions are important to the development of progressive sophistication in skills within a domain. One view of learning progressions suggests that they are presented to students as a continuum of learning, accounting for different rates of learning (DeMeester & Jones, 2009). The rate of individual student‘s progress may vary along the learning progressions, but progressions should ultimately connect the knowledge, concepts, and skills students develop as they evolve from novice to more expert performances (Heritage, 2008). In this way, teachers and students should be able to ―see and understand the scaffolding they will be climbing as they approach‖ (Stiggins, 2005, p. 327) learning goals. In addition, if learning is derailed at any point, a teacher can identify this and adjust accordingly.

It is fair to say that if the standards do not present clear descriptions of how students learning progresses in a domain, then they are unlikely to be useful for formative assessment. Standards are insufficiently clear about how learning develops for teachers to be able to map formative assessment opportunities to them. This means that teachers are not able determine where student learning lies on a continuum, and know what to do to close the gap between current learning and desired goals. Explicit learning progressions can provide the clarity that teachers need. By describing a pathway of learning they can assist teachers to plan instruction. Formative assessment can be tied to learning goals and the evidence elicited can determine students’ understanding and skill at a given point. When teachers understand the continuum of learning in a domain and have information about current status relative to learning goals (rather than to the activity they have designed to help students meet the goal), they are better able to make decisions about what the next steps in learning should be.

This progression of learning allows teachers to position their students' learning, not only in relation to their current class(es) and the objectives for that cohort, but also in relation to prior and subsequent classes. Consequently, teachers are able to view current learning against a bigger picture of development. In terms of instruction, they are able to make connections between prior and successive learning. Also, information from formative assessment can be used to pinpoint where students’ learning lies on the continuum. Sometimes this will mean that teachers have to move backwards along the continuum, for example, if key building blocks are missing. Similarly, they might move learning further forward if some students are outpacing their peers. In both cases, the continuum allows them to make an appropriate match between instruction and the learners' needs.

Assessment for learning - Formative Assessment

Assessment for learning was originally conceived of as formative assessment and placed in contrast to summative assessment. Michael Scriven proposed the terms formative and summative in 1967 to explain two distinct roles that evaluation could play in evaluating curriculum. In the years to follow, Benjamin Bloom and colleagues (1969; 1971) suggested applying the same distinction to the evaluation of student learning— ―what we tend today to call assessment‖ (Wiliam, 2006, p. 283). Subsequently, the terms formative and summative have become fundamental to understanding assessment in education. Summative assessment focuses on summing up or summarizing achievement of students, classes, schools, etc. (Bloom, Hastings, & Madus, 1971; National Research Council [NRC], 2001; Sadler, 1989; Shavelson, 2006). Formative assessment centers on active feedback loops that assist learning (Black & Wiliam, 2004; Sadler, 1989; Shavelson, 2006). Recently, some scholars have begun to refer to summative assessment as assessment of learning and formative assessment as assessment for learning (Black & Wiliam, 2003; Broadfoot, 2008; Gipps & Stobart, 1997; Stiggins, 2002).

In the years since Scriven‘s identification and Bloom‘s extension of summative and formative assessment types, ―the interest (and investment) in summative assessment has far outstripped that accorded to formative assessment‖ (Stiggins, 2005, p. 326). Black and Wiliam (2003) discuss in some detail the ups and downs of formative assessment during the 1970s through the late 1980s. In the late 1980s, two substantial review articles (Crooks, 1988; Natriello, 1987) and a seminal piece on the function of formative assessment in the development of expertise (Sadler, 1989) boosted interest in assessment for learning. This growing interest appeared to be substantiated with Fuchs and Fuchs‘ (1986) meta-analysis and Black and Wiliam‘s (1998) comprehensive review of about 250 articles. Both studies reported significantly positive student learning gains. The Black and Wiliam (1998) work demonstrated gains of a half to a full standard deviation, with low-achieving students making the largest increase. Although Dunn and Mulvenon (2009) have recently contested the conclusiveness of these two studies,1 other recent research has also shown positive impact on student learning (e.g., Black, Harrison, Lee, Marshall, & Wiliam, 2004; Ruiz- Primo & Furtak, 2006). Moreover, Dunn and Mulvenon (2009) were unable to present any examples of formative assessment producing negative achievement results.

In addition to the consensus emerging around the potential benefits of formative assessment practice, scholars generally agree that formative assessment is the process of using information about students‘ learning on the course of instruction to make decisions to improve learning (Atkin, Black, & Coffey, 2001; Bell & Cowie, 2001; Black, 1993; Black, Harrison, Lee, Marshall, & Wiliam, 2003; Black & Wiliam, 1998, 2004; Harlen, Gipps, Broadfoot, & Nuttall, 1992; Harlen & James, 1996; Tunstall & Gipps, 1996; Shepard, 2000).

How the process of formative assessment is conceptualized and implemented still varies somewhat, but all of the researchers listed above agree that regular testing and simply informing students of their scores does not constitute formative assessment. Instead, according to Black and associates (2004), the evidence of student understanding (and learning) evoked from one round of the formative assessment process should be ―used to adapt the teaching work to meet learning needs‖ (p. 2). In 2007, Formative Assessment for Students and Teachers State Collaborative (FAST SCASS) of the Council of Chief State Officers with national and international researchers in formative assessment identified five attributes of the formative assessment process from the literature. They are as follows:

 Learning progressions should clearly articulate the sub-goals of the ultimate learning goal.

 Learning goals and criteria for success should be clearly identified and communicated to students.

 Students should be provided with evidence-based feedback that is linked to the intended instructional outcomes and criteria for success.

 Both self- and peer-assessment are important for providing students an opportunity to think metacognitively about their learning.

 A classroom culture in which teachers and students are partners in learning should be established. Margaret Heritage (2007) of the National Center for Research on Evaluation, Standards, and Student Testing (CRESST) folded the attributes into a model of the formative assessment process that is applicable . The process focuses the work on the following four elements of formative assessment: learning progressions, including learning goals and success criteria, identifying the gap, eliciting evidence of learning, teacher assessment, teacher feedback, and student involvement.

COGNITIVE TRAITS - Reasoning Ability

With respect to reasoning abilities, we can distinguish between inductive, deductive and abductive reasoning. In the following discussion, we will focus on inductive reasoning, since this ability is the most important one regarding learning. We shall also provide some discussion on deductive reasoning. Inductive reasoning skills relate to the ability to construct concepts from examples. When a student faces a complicated problem, he/she looks for known patterns, and uses them to construct a temporary internal hypotheses or schema in which to work (Bower & Hilgard, 1981). It is easier for students who possess better inductive reasoning skill to recognize a previously known pattern and generalize higher-order rules. As a result, the load on working memory is reduced, and the learning process is more efficient. In other words, the higher the inductive reasoning ability, the easier it is to build up the mental model of the information learned. According to Harverty, Koedinger, Klahr, & Alibali (2000) inductive reasoning ability is the best predictor for academic performance. For simulation based discovery learning, students are asked to infer characteristics of a model through experimentations by using a computer simulation and thus are asked to use their inductive reasoning skills. According to Veermans and van Joolingen (1998) simulation based discovery learning results in deeper rooting of  the knowledge, enhanced transfer, the acquisition of regulatory skills, and would yield better motivation. However, discovery learning does not always yield to better learning results. One of the reasons is that students have difficulties in performing the required processes. To improve the learning progress and support learners with low inductive reasoning abilities, Veermans and van Joolingen have designed a mechanism that provides advices based on the performed experiments in the simulation. This mechanism is integrated in SimQuest, an authoring system for simulation based discovery (van Joolingen & de Jong, 2003).

Considering again exploratory learning and the Exploration Space Control elements, for learners with low inductive reasoning skills, many opportunities for observation should be provided. Therefore, learning systems can support these learners by providing a high amount of well-structured and concrete information

with many paths. For learners with high inductive reasoning skills, the amount of information and paths should decrease to reduce the complexity of the hyperspace and hence enable the learners to grasp the concepts quicker. Moreover, information can be presented in a more abstract way (Kinshuk & Lin, 2003). Deduction is defined as drawing logical consequences from premises. An application for deductive reasoning is, for instance, naturalistic decision making (Zsambok & Klein, 1997), which deals with examining what people do in real world situations. It has been observed that experienced decision makers recognize the situation and associate an appropriate solution whereas unexperienced decision makers perform an unorganized and almost random search of alternatives. When it comes to complex situations, humans often fail in finding appropriate solutions. According to Dörner (1997) several reasons exist for such failures, for example, humans tend to oversimplify the mental model of the complex system, tend to be slow in thinking when it comes to conscious thoughts, or tend to ignore the possibility of side-effects. However, Dörner’s experiments showed that leaders from business and industry tend to make more effective decisions in complex situations. Therefore, he argued that the necessary behavior and skills can be acquired and learnt.

NUMERACY ACROSS ALL LEARNING AREAS

Numerate behavior can be thought of as the disposition and competence to use mathematics in the service of endeavors other than mathematics. Numeracy is linked to ‘what mathematics you know’, but it also involves the skills, thinking processes and attitudes needed to choose and use mathematics outside mathematics. In this sense, numeracy is about practical knowledge that has its origins and importance in the physical or social world rather than in the conceptual field of mathematics itself.

The major responsibility for developing students’ numeracy lies with the Mathematics learning area. At all levels, teachers of mathematics should help students learn to use their mathematics to solve practical problems and as a tool for learning beyond the mathematics classroom. Teachers of mathematics should also take responsibility for teaching students to read and write in situations that involve mathematical ideas, notations and visual forms.

However, the development of numeracy involves more than mathematics classrooms alone can provide, since its achievement requires working mathematically in a range of different settings. Indeed, school mathematics is unable to fully capture all that is numeracy simply because the mathematics is in mathematics. In order for them to be ‘numerate’, students must learn to connect the mathematics from situation to situation

– including across the school curriculum and beyond. Learning areas other than mathematics can, therefore, contribute to the enhancement of students’ numeracy by:

• providing rich contexts in which students can use their mathematics;

• expecting students to use their mathematics in other learning areas; and

• maintaining common and challenging standards. 

In The Arts learning Area, for example, students might consider what shapes they and others find pleasing. In doing so they are likely to use and also to enrich the language of shape and number developed in mathematics. Mathematics lessons might build upon this work on form, using the golden ratio as a starting point for considering ratios in general. Mathematics lessons could also involve students in trying to find out whether people actually do prefer particular shapes and forms, thus developing important ideas about data collection and handling and about the value and applicability of their mathematical work.

The Society and Environment and the Health and Physical Education learning areas may each call upon and enrich number, measurement and data skills when students investigate such matters as water wastage or rubbish generation and disposal in the school yard, their own fitness levels or adolescent health. The Society and Environment and Mathematics learning areas may also provide complementary work which develops an understanding of how three-dimensional space can be represented in two dimensions – as in various map projections – but never without some distortion, so that interpreting a two-dimensional representation requires an understanding of the features of the three-dimensional space that are and are not preserved in the two-dimensional representation.

The Technology and Enterprise learning area will also both draw upon and enhance students’ understanding of the representation of three-dimensional space in two dimensions. Indeed, the design and production of models involved in achieving the outcomes of Technology and Enterprise will draw extensively upon number,

measurement and space concepts and skills, and should also considerably enhance students’ learning in these areas by providing a broader range of contexts and experiences than could be provided in mathematics alone.

The Science, Society and Environment and Mathematics learning areas each contribute to students’ understanding of how time may be represented sequentially or in a linear fashion and of how the tracking and measurement of time relates to cyclical or periodic phenomena. These learning areas, together with the Languages Other than English learning area, can help students to understand that people may conceive of time differently and in culturally-specific ways. Thus, students can come to understand that the application of particular mathematical ideas to the measurement and recording of time is both influenced by and influences how we think about and experience time in our daily lives and in our myths and legends. The English learning area provides the language foundations essential for the learning of mathematics and the development of numeracy. Equally, developing students’ capacities to draw on a wide range of mathematical ideas in their reading and viewing generally is a major contribution of numeracy to English. English and mathematics together provide the basic information skills involved in reading the daily newspaper or a telephone book, and in preparing reports.

Thus, within each learning area, two questions should be asked:

• How can the learning area enhance students’ numeracy?

• How can numeracy contribute to enhanced outcomes in the learning area?

Answering these two question will require teachers to collaborate in developing common interpretations of numeracy and strategies to assist their students to use mathematics across and beyond the school curriculum.

LINKS TO THE OUTCOMES IN THE OVERARCHING STATEMENT - MATHS LEARNING

Mathematics is integral in the education of young people. The Mathematics learning area, either directly or indirectly, contributes to each of the Overarching outcomes. In addition, through numeracy, it provides learning skills which contribute the achievement of the outcomes for most of the learning areas of the curriculum.

Students’ achievement of the outcomes in Working Mathematically, Number, Measurement, Chance and Data, Space, and Algebra will enable them to deal with quantitative and spatial ideas; to collect, display, analyse and interpret data; and to describe and reason about patterns and relationships. Thus, the Mathematics learning area takes a major – although certainly not sole – responsibility for ensuring that

students: 

• select, integrate and apply numerical and spatial concepts and techniques;

• recognise when and what information is needed, locate and obtain it from a range of sources and evaluate, use and share it with others; and

• describe and reason about patterns, structures and relationships in order to understand, interpret, justify and make predictions.

The Mathematics learning area supports and reinforces the literacy work of the school by setting expectations and providing feedback and support to students that is consistent with current thinking in literacy. It also takes a major responsibility for assisting students to learn to read, write, listen to and talk about mathematics, and to develop the range of special symbols, vocabulary and diagrammatic representations that mathematics contributes to language. In these ways, the Mathematics learning area makes a direct contribution to enabling students to: 

• use language to understand, develop and convey ideas and information and interact with others.

Mathematics plays a central role in the generation of technology generally and has, itself, changed significantly as a result of the impact of computing technologies. Through the achievement of a number of the Mathematics learning area outcomes, students:

• select, use and adapt technologies.

Through the Mathematics learning area, students come to appreciate the way in which mathematics is embedded in the very fabric of our own and other societies. Their understanding of the cultural and intellectual significance of mathematical activity, and how it influences what we are and might be, enhances the extent to which they:

• understand their cultural, geographic and historical contexts and have the knowledge, skills and values necessary for active participation ;

• interact with people and cultures other than their own and are equipped to contribute to the global community; and

• participate in creative activity of their own and understand and engage with the artistic, cultural and intellectual work of others.

The outcomes in Appreciating Mathematics and Working Mathematically address the mathematical attitudes, appreciations, and individual and collaborative work habits that enable students to be critical, creative and confident users of mathematics. With the provision of a supportive environment for learning mathematics, appropriate mathematical challenge and teaching processes that foster autonomous learning in mathematics, this means that students:

• visualise consequences, think laterally, recognise opportunity and potential and are prepared to test options;

• are self-motivated and confident in their approach to learning and are able to work individually and collaboratively; and

• recognise that each person has the right to feel valued and be safe and, in this regard, understand their rights and obligations and behave responsibly.

Finally, the Mathematics learning area makes an indirect contribution to several Overarching outcomes – and hence to other learning areas – as a result of its responsibilities for numeracy. Numeracy generally benefits learning in most parts of the school curriculum, while being innumerate can inhibit and even prevent student

learning elsewhere. In particular, being numerate can significantly enhance students’ capacity to:

• understand and appreciate the physical, biological and technological world and have the knowledge and skills to make decisions in relation to it;

• understand their cultural, geographic and historical contexts and have the knowledge, skills and values necessary for active participation ;

• value and implement practices that promote personal growth and well-being; and

• participate in creative activity of their own and understand and engage with the artistic, cultural and intellectual work of others.

Issues in the design of technology-supported learning environments utilizing cognitive conflict

The design and use of learning environments utilizing cognitive conflict to promote learning necessarily involve consideration of three processes: concept representation, conflict recognition and conflict resolution. Of these, the process most critical for design is, concept representation. The effectiveness of the learning environment in facilitating the learner's recognition and resolution of cognitive conflict depends largely on effective concept representation in the environment.

In the confrontational approach the 'correct' or target structure must be represented in the environment. Typically in this approach the designer will make assumptions about the cognitive structures likely to be brought to the situation by learners but will not attempt to represent these directly in the environment. The guiding approach requires that the designer have a good appreciation of the conceptual structures likely to be brought to the situation by learners, as this approach requires the planning of experiences that will help learners to build on their existing conceptual structures to develop more 'appropriate' ones. Explicit representation of these structures in the environment may not be necessary. However if the developmental sequence is to be effective, the designer must apprehend not only learners' initial conceptual structures but also changes to these that are likely to result from experiences within the environment.

From the point of view of design, the explanatory approach is perhaps the most subtle of the three described. In comparison with the previous approaches, this one requires the designer to have a clear understanding of the conceptual structures of the learner and, further, a reasonable idea of the learner's underlying rationale for these structures. In the ball toss example, the designer is confident that learners can make correct predictions about the motion based on their current conceptual structures, but he anticipates that these structures contain weaknesses that are revealed only when reasons or explanations for the

predictions are investigated.

Through increasing the options for concept representation, the use of technology has enhanced considerably the potential for design and development of learning environments utilizing cognitive conflict. An exhaustive discussion of the possibilities here is beyond the scope of the present paper. However it is appropriate to mention the ability of these environments to provide multiple formalisms for representation of concepts for students, including linguistic, graphical, animated, mathematical, programming code and many other representational modes.

For effective utilization of cognitive conflict in a learning environment, the designer must ensure that the consequences of the differences between 'target' and learner conceptual structures are evident and understandable by the learner. For learners, apprehension and articulation of a puzzling or unexpected situation is a major step towards resolving it. In environments designed with the confrontational approach the conflict is generally immediate and very obvious. In the guided approach where the designer expects gradual,

probably small changes in .the learners' cognitive structures, the recognition of conflict may not be as striking.

Using the explanatory approach the designer must take great care to ensure that the recognition of conflict with previously held understandings in fact actually happens. This might mean directing the learner's attention quite deliberately to the aspects of the environment designed for this, perhaps through suggested activities to be undertaken in the environment or by some other suitable design strategy.

Some aspects of technology-supported learning environments can be used to enhance the processes of conflict recognition and resolution. For example, interactivity in well designed technology-based learning environments can provide feedback to learners about the appropriateness of their assumptions and predictions as they work in the environment, and technology might be used to sponsor collaboration to help conflict resolution.

The processes of concept representation, conflict recognition and conflict resolution are discussed, and the most important of these from a design perspective, concept representation, is examined in some detail in the context of the three approaches. Finally to considers the power of technology-supported learning environments to enable, particularly through increased options for concept representation, the design and development of learning environments that utilize cognitive conflict in the more complex and subtle ways described in the three approaches delineated in the article.Although much of the recent work on cognitive conflict has been undertaken in education, the argument presented in the article should not be confined to learning of particular subject area . Papert points out that 'Piaget has shown that children hold false theories as a necessary part of the process of learning to think' (Papert, 1980: 132-3). It is contended that the argument made here has application learning in a wide variety of subject areas and contexts.

Cognitive conflict and learning

Piaget used the term disequilibration to describe the process of an individual's encountering a new experience that generates a contradiction with the individual's existing cognitive structures (see for example Piaget, 1977), and argued that cognitive development or learning occurs as the individual attempts to resolve this cognitive conflict, a process he referred to as accommodation. He proposed three possible types of accommodation: the individual might ignore the contradiction; the individual might hold two theories simultaneously, dealing with the contradiction by applying one theory in some specific cases, and the other in others; or the individual might construct a new, more encompassing notion that explains and resolves the prior contradiction. Achieving this last type of accommodation, that is learning, is the purpose of teaching and of educational software design in the present context.

Some other writers, a leader among whom is Seymour Papert, regard the issue of cognitive conflict with a somewhat different emphasis from that in the body of work just described. Papert considers cognitive conflict to be part of the vital process of theory building in learning. He assigns rather less importance to the efficient achievement of 'correct' concepts, valuing the unorthodox theories and explanations developed by learners. He emphasizes the importance for learning of this process of building and modification of theories.

Papert argues that the unorthodox theories of young children are not deficiencies or cognitive gaps, but that they serve as a way of flexing cognitive muscles, or developing and working through the necessary skills needed for more orthodox theorizing. He asserts that educators distort Piaget's message by seeing his contribution as revealing that children hold false beliefs, which they, the educators, must overcome (Papert, 1980). This leads to the development of learning environments that use cognitive conflict somewhat less directly than implied by the science education work outlined earlier. It suggests the creation of open-ended learning environments in which students might explore concepts and ideas, developing (possibly 'wrong' or transitional) theories and testing these. Papert uses the word microworlds to describe such learning environments.

Microworlds (Papert, 1980; Squires and McDougall, 1986) are self-contained environments, simple restricted worlds 'in which certain questions are relevant and others are not' (Papert, 1980: 117). Learners explore the properties of a chosen microworld in a completely open-ended fashion, developing and testing theories about the environment as they do so. Papert advocates the construction of many such microworlds, each with its own set of assumptions and constraints; a microworld designer can lead a learner to new understandings by careful control of these assumptions and constraints. In later work (see for example Harel and Papert, 1991; Harel, 1991) Papert and his colleagues advocate the construction by learners of microworlds of their own; however, this paper is concerned only with the development for teaching purposes of environments designed specifically to promote cognitive conflict in learners.

Andrea diSessa, who worked for some years with Papert, assigns even greater importance to learners' pre-existing cognitive structures. He discusses these in terms of intuitive knowledge, referring to the many fragmentary small structures that a learner gleans from experience in the world as phenomenological primitives or p-prims (diSessa, 2000). DiSessa describes cognitive conflict in the following way. People have hundreds if not thousands of p-prims, and when they make a judgement of reasonableness or unreasonableness they are frequently summoning all the relevant p-prims and deciding which one, or which collection of a few, best matches the situation. Then what happens in the situation is reasonable if it matches the chosen p-prim or p-prims and surprising if it does not. Like Papert, diSessa is critical of what he sees as educators' under-valuing of learners' intuitive knowledge, arguing that learners are adjusting their p-prims to new experiences all the time and that it is not difficult to make small changes of this kind.

Problem based learning method/model

In the problem based learning model the students’ turn from passive listeners of information receivers to active, free self-learner and problem solvers. It also shifts the emphasis of educational programs from teaching to learning. It enables the students to learn new knowledge by facing the problems to be solved instead of feeling boredom. Problem based learning affect positively certain other attributes such as problem solving, information acquisition, and information sharing with others, group works, and communication etc. Again problems solving is a deliberate and serious act, involves the use of some novel method, higher thinking and systematic planned steps for the acquisition set goals. The basic and foremost aim of this learning model is acquisition of such information which based on facts (Yuzhi, 2003 & Mangle, 2008).

According to Gallagher et. al,(1999) in problem based learning environment, students act as professionals and are confronted with problems that require clearly defining and well structured problems, developing hypothesis, assessing, analyzing, utilizing data from different sources, revising initial hypothesis as the data collected developing and justifying solutions based on evidence and reasoning. The practice of problem based learning is richly diverse as educators around the world and in a wide range of disciplined have discovered it as a route to innovating education, The educators used problem solving method as an educational tool to enhance learning as a relevant and practical experience, to have students’ problem solving skills and to promote students’ independent learning skill. Eng (2001) opined problem based learning as a philosophy aims to design and deliver a total learning environment that is holistic to student-centered and student empowerment.

Presenting the students with a problem, give them opportunity to take risks, to adopt new understandings, to apply knowledge, to work in context and to enjoy the thrill of being discoverers. Tick, (2007) stated that in the student-centered learning environment that is desirable for problem based learning, the central figure of the learning-teaching process is the student. The learning objective is not the reproduction, recall and learning of passively received learning material but the active and creative engagement of students in group work and in individual study thus transferring the skills and knowledge. The individual, autonomous self-directed learning gives the freedom to the learner to decide individually and consciously on the learning strategy and on the time scale, s/he wants to follow.

The most important achievement of a teacher is to help his/her students along the road to independent learning. In problem based learning, teacher acts just as facilitator, rather than a primary source of information or dispenser of knowledge. Roh, (2003) argued that within problem based learning environments, teachers' instructional abilities are more critical than in the traditional teacher-centered classrooms. Beyond presenting mathematical knowledge to the students, teachers in problem based learning environments must engage students in marshaling information and using their knowledge in applied sand real settings. Evidence of poor performance in mathematics by elementary school students highlight the facts that the most desired technological, scientific and business application for mathematics cannot be sustained. This makes it paramount to seek for a strategy for teaching mathematics that aims at improving its understanding and performance by students practically (Okigbo & Osuafor, 2008). Problem solving as a method of teaching may be used to accomplish the instructional roles of learning basic facts, concepts, and procedure, as well as goals for problem solving. Problem solving is a major part of Mathematics has many applications and often those applications represent important problems in, mathematics. We include problem solving in school mathematics because it can stimulate the interest and enthusiasm of the students (Wilson, 1993).

PROBLEM-BASED LEARNING AND MATHEMATICS

Starting from the assumption that problem-based learning is an instructional approach suitable for transmitting

a real understanding of mathematics, we have to regret that such a methodology is not common in real teaching practice at school, as teachers usually rely on self perpetuating “traditional” methods. These  include:

• training courses for pre-service and in-service teachers;

• organizing problem-based mathematical laboratories in school (both in primary school and at a higher level);

• designing interactive exhibitions;

• organizing web-based mathematical game contests;

• publishing books and magazines.

It is my believe that the role of Research is crucial in carrying on synergical actions both on the teachers and on the pupils in order to improve the effectiveness of the teaching of mathematics.

Problem-based learning (PBL) is a constructivist learner-centred instructional approach based on the analysis, resolution and discussion of a given problem. It can be applied to any subject, indeed it is especially useful for the teaching of mathematics . For a neat definition of PBL we refer to according to which PBL “is an instructional (and curricular) learner-centered approach that empowers learners to conduct research, integrate theory and practice, and apply knowledge and skills to develop a viable solution to a defined problem”. Typically a PBL session follows these steps :

• pupils are given a problem;

• they discuss the problem and/or work on the problem in small groups, collecting information useful to solve the problem;

• all the pupils gather together to compare findings and/or discuss conclusions; new problems could arise from this discussion, in this case

• pupils go back to work on the new problems, and the cycle starts again.

In spite of researches documenting the effectiveness of PBL, we have to regret that such a methodology is not common in real teaching practice at school, as teachers usually rely on self perpetuating “traditional” methods . Research should thus focus on finding ways to support teachers practice in schools.

When designing refresher courses for teachers, it is important to focus not only on disciplinary contents, but also on methodological aspects. That is, on one side one has to deal with the peculiar nature of mathematics, keeping in mind that ordinary people usually have a conflicting relation with mathematics. This bad attitudes is often due to wrong beliefs about the subject, which is seen as a dull set of rules to be learned by heart. Thus teachers should give their students the opportunity to see mathematics as a living subject, that is as an exploratory, dynamic, and evolving science. Teachers should teach their students that the short answer is not what they really need, and that things must be looked at in depth. And this is the place where methodology steps in: simply put, if mathematics is an exploratory subject, then it must be taught in an exploratory way. The socio-constructivist perspective enforce this point of view, and doing mathematics in a PBL learner centred environment is a chance to give students a real understanding of the subject. Once again, the basic idea is that you do not appreciate and learn mathematics if you do not do mathematics.

Moreover, once accepted that PBL is an effective way to teach mathematics, one has to act on teachers’ beliefs in order to have them actually use PBL in their teaching practice. According to my experience, the most effective way to have teacher willing to use a PBL methodology is to have they experience themselves the joy of mathematical discovery, hence the courses for in-service teachers now adopt PBL. By undergoing PBL, teachers acquire beliefs in favour of  a PBL methodology as they see that in this way they learn some good maths . The main criticism concerns the fact that PBL is time consuming, and it is thus important to show teachers that the time is well spent as pupils get a real understanding of the subject, and gain at the same time many collateral competencies . A further objection coming from the teacher is that such activities rely on “good” problems and in-service teachers do not feel to be competent enough to create suitable ones. Researchers can provide teachers with ready to use material. There is one more aspects that needs to be stressed in order to use PBL teachers must be trained as their role is crucial. Once more the personal experience is the best ground on which they can build an effective teaching practice.

Basics of Set Theory - A breif perspective

When you start reading this, the first thing you should be asking yourselves is “What is Set Theory and why is it relevant?” Though Propositional Logic will prove a useful tool to describe certain aspects of meaning, like the reasoning in (1), it is a blunt instrument.

(1) a. Jacques sang and danced

Therefore: Jacques sang

b. If Jacques kisses Smurfette, she will slap Jacques

   Jacques kissed Smurfette

Therefore: Smurfette slaps Jacques

Propositional Logic won’t help us with the argument in (2), which is why we will extend it to Predicate Logic.

(2) Every man is mortal

     Aristotle is a man

Therefore: Aristotle is mortal.

Our intuitions tell us that if the first and second sentence are true, then the third necessarily follows. The reasoning goes like this. Suppose the first sentence is true: the collection of men is part of the collection of mortal things. Now suppose the second sentence is also true: Aristotle is a man. The first sentence says that if a thing is in the collection of men, it is also in the collection of mortal things. So it follows that Aristotle is a mortal thing (or simply mortal), which is what the third sentence states.  The semantics of Predicate Logic is defined in terms of Set Theory.

A bunch of carrots, a classroom full of students, a herd of elephants: these are all examples of sets of things. Intuitively, you can think of a set as an abstract collection of objects, which may correspond to things in the world or to concepts, etc. A set is sometimes also called a collection.1 The objects that are collected in a set

are called its members or elements. So, a carrot, a student, and an elephant, are members of the sets described above. A Set may also contain other sets as its members.

I offer no definition of what a set is beyond the intuitive notion described above. Instead, I am going to show you what can be done with sets. This is a typical approach to Set Theory, i.e., sets are treated as primitive s of the theory and are not definable in more basic terms. I adopt the notation in (4) for convenience.

(4) a. Capital letters represent sets: A, B, C, …

     b. Lower case letters represent members of sets: a, b, c, …, sometimes x, y, z, … The principal concept of Set Theory is belonging, i.e., elements belonging to sets. And this concept is represented using the membership relation, expressed by the rounded Greek letter epsilon ∈, as in (5). It’s negation is expressed using a barred epsilon ∉ as in (6).

(5) x ∈ A

x is an element of the set A.

(6) x ∉ A

x is not an element of the set A.

Elements

The objects in sets are called elements (or members).

If an element e belongs to set S this is represented as e∈S.

If an element f does not belong to set S this is represented as f∉S.

Writing sets

1. Listing method

All of the elements of a set are written.

A = {1, 3, 5, 7, 9}

2. Set - builder method

A typical element is named, along with its description, such as:

B = {x | x is an odd number from 1 to 10}

The vertical bar | is read: ‘‘such that’’, and it also can be written as :

3. Venn diagram

A Venn diagram is used for graphical representation of the set as a collection of its objects.

Example 1

This is a typical set, consisted of numbers 1, 2 and 3 listed inside of braces:

{1, 2, 3}

Alternatively we may choose to describe this set like this

{1, 2, 3} = { integers greater than 0 and less than 4 }

More formally, we may use this notation:

{1, 2, 3} = { x : x is an integer and 0 < x < 4} = { x : x  Z, 0 < x < 4 }

Singleton set

Singleton set is a set with only one element.

A singleton set is different from the element itself and it is denoted by the element with surrounding braces.

Ex. 5 – this is number 5

{5} –this is singleton set containing number 5

Empty Set

A set with no elements in it is called an empty set.

This set is unique and it is denoted with the symbol Ø or with empty braces { }.

Infinite set

Set which contain an infinite number of elements is called infinite set.

These sets are often given by a few elements and ellipses which indicate that this sequence continues indefinitely.

A = {1, 2, 3, … }

Subsets

If all the elements of one set X are also elements of another set Y, then X is said to be a subset of Y. Similarly  if elements of Z are not elements of another set Y so it is not a subset of Y.

Superset

If a set X is a subset of Y, then Y is said to be a superset of X. Every set is its own subset and superset. The empty set is a subset of all sets.

Example 2

Let A = {1, 2, 3} and B = {1, 2}.

B is a subset of A.

A is not a subset of B then  notice that: n (B) < n (A).

SET OPERATIONS

Example: Let A = {10, 11, 12, 13} and B = {12, 13, 14}.

Intersection

Symbol A ∩ B denotes the intersection of two sets.

It is a set comprised of the elements which are in the both sets. We can write this as:

A ∩ B = { x| x∈  A and x ∈ B }

In the example A ∩ B = {12, 13}

If two sets have no elements in common, i.e. A ∩B = Ø we say that they are disjoint.

Union

Symbol A U B denotes the union of two sets.

It is a set comprised of the elements which are in the either set (at least one). We write this as:

A U B = { x : x ∈ A or x ∈ B }

In the example A U B = {10, 11, 12, 13, 14}

Universal set

The set containing all elements of a problem is called the universal set (ℰ).

That is a set from which elements may be considered.

(The notation comes from the French word ensemble).

Complement

If ℰ is universal set and A is a subset of ℰ, then the complement of a set A, denoted by A’, is the

set of all elements not in A.

A’ = { x : x ∉ A }

Cardinality of sets (Number of elements)

The number of elements in a set A is called its cardinality and is denoted by n (A)

Example. If A = {a, b, c, d, e, f} then n (A) = 6.

Properties of Cardinality:

1. n(Ø) = 0

2. n(A’) = n(ℰ) – n(A)

3. if A ∩ B ≠ Ø then

n(AUB) = n(A) + n(B) – n(A ∩ B)

4. if A ∩ B = Ø (disjoint sets) then

n(AUB) = n(A) + n(B)

Set-theoretic equalities

There are a number of general laws about sets which follow from the definitions of settheoretic

operations, subsets, etc. A useful selection of these is shown below. They are

grouped under their traditional names. These equations below hold for any sets X, Y, Z:

1. Idempotent Laws

(a) X ∪ X = X (b) X ∩ X = X

2. Commutative Laws

(a) X ∪ Y = Y ∪ X (b) X ∩ Y = Y ∩ X

3. Associative Laws

(a) (X ∪ Y) ∪ Z = X ∪ (Y ∪ Z) (b) (X ∩ Y) ∩Z = X ∩ (Y ∩ Z)

4. Distributive Laws

(a) X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z) (b) X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z)

5. Identity Laws

(a) X ∪ ∅ = X (c) X ∩ ∅ = ∅

(b) X ∪ U = U (d) X ∩ U = X

6. Complement Laws

(a) X ∪ X’ = U (c) X ∩ X’ = ∅

(b) (X’)’ = X (d) X – Y = X ∩ Y’

7. DeMorgan’s Laws

(a) (X ∪ Y)’ = X’ ∩ Y’ (b) (X ∩ Y)’ = X’ ∪ Y’

8. Consistency Principle

(a) X ⊆ Y iff X ∪ Y = Y (b) X ⊆ Y iff X ∩ Y = X