Knowing What You Know: Assessment Competence

Learning is any process that enables a change in a person’s capacity to understand themselves and the world around them. Assessment is any feedback that enables a learner to know when there has been a change in their capacity to understand themselves and the world around them. Learning and assessment thus occur in the course of daily life as individuals interact with their environments, with each other, and as they internalize new knowledge and new ways of understanding. Assessment is integral to learning, whether provided in a formal classroom setting or any other context.

But learning is also an intentional process and competences for “learning to learn” are considered vital for effective lifelong learning.1 Learning-to-learn involves awareness of how and why an individual acquires and processes different types of knowledge (meta-cognition), and the learning methods, environments and habits she or he finds to be most effective (self-regulation) (Eurydice, 2002). Assessment is integral to all forms learning, whether the apparently effortless learning that occurs through interactions in daily life, or through intentional learning.

Formal assessments are based on quantitative and/or qualitative measurements. For novice learners, assessments are usually based on progress toward specific goals with clear criteria. At higher levels, learners may be working individually or collectively to solve problems with no clearly defined outcome, and assessment is part of this exploratory process. Feedback is an essential element in assessment to improve learning. Feedback may be provided in the course of an interactive discussion, in writing, in the context of a game, or through signals in the learner’s environment. It is most effective when it is timely and at an appropriate level of detail to suggest next steps (Black and Wiliam, 1998). In the case of goal-directed learning, feedback needs to be congruent with the learning aims. When learning is exploratory and open-ended feedback may, in some cases, be adequately conveyed through conventional and familiar indicators or it may not. In the latter case inventing new indicators and becoming sensitive to the meaning of such feedback becomes one of the key parts of improving assessment.

As already noted the context and technologies of learning and assessment are changing rapidly. Part of this change calls into question how we define and achieve the competences for learning to learn. Indeed, the term “competence”, which refers to the ability to mobilize both cognitive and non-cognitive resources in unknown contexts, seems particularly apt here. Learning, in both open and closed systems, requires learners to define what, how, with whom and why they learn; they and their collaborators, mentors and teachers need to extend and deepen their knowledge of effective assessment processes, and their ability to interpret and respond to feedback. Today’s changing context is calling for a deeper understanding of the factors (rational and non-rational) that enhance the ability of learners in all situations and environments to identify and track changes in their “capacity to understand themselves and the world around them”. This is assessment competency. 

New technologies are changing learning and assessment – and raising learners’ expectations that they will have a say in what and how they learn. They may also support new approaches to learning and education:

• Social media facilitate information sharing, usergenerated content, and collaboration in virtual communities. Increasingly, individuals expect the opportunity to participate in knowledge creation and to share their own assessments through blogs, wikis, feedback and rating services. Those with special interests may engage in collective learning and problem solving on a collective (crowd sourced) scale. In the education world, Open Educational Resources allow educators to share and shape learning content, and to provide input on effectiveness.

• Tracking programmes and electronic portfolios allow individuals to monitor their performance toward goals

(criterion-referenced assessments), to compare their progress against their own prior performances (ipsativereferenced assessments), or against the progress of other users (norm-referenced assessments). For example, Baumeister and Tierney (2011) note that several online programmes that help individuals to track exercise, sleep and savings habits can be very effective in supporting goal achievement. The research suggests that individuals who use tracking programmes to focus on how much further  they have to go (rather than progress already made) (Fishbach and Koo, 2010), and who share their progress with others are more likely to succeed in reaching goals.

• Video games both entertain players and provide rapid feedback on progress. Many games step up challenges as players develop their skills. Educationalists are now working with game developers to create games focused on building specific competences. For example, Popoviæ describes how novice learners in biochemistry have engaged in collective game play to address specific puzzles and problems within the discipline, and in the process have found solutions that have eluded experts. Delacruz and colleagues (2010) have found that effectively-designed games can tap into mathematical thinking and can be used for summative assessments of learning, and potentially, as more data on players’ thinking processes are gathered through game play, as a tool for formative assessment.

• Test developers are also focusing on how ICT-based assessments may support the integration of large-scale, external summative assessments and classroom-based formative assessments. Currently, there are a number of technical barriers to this kind of integration. Data gathered in large-scale assessments do not provide the level of detail needed to diagnose individual student learning needs, nor are they delivered in a timely enough manner to have an impact on the learning of students tested. There are also challenges related to creating reliable measures of higher-order skills, such as problem solving and collaboration.

• Learning Analytics, as discussed by Siemens in the following section, and by Shum in his work on Social Learning Analytics, take these ambitions a step further. In a previous paper, Siemens and Long (2011) noted

that social network analysis tracking learners’ online behaviour could potentially provide detailed information (“big data”) on learning processes and real time assessment of progress, along with suggestions for next steps. Importantly, learning analytics may also support “intelligent curriculum” and “adaptive content”, allowing deep personalisation of learning. Shum and Crick (forthcoming 2012) propose learning analytics as a tool to assess learners’ dispositions, values, attitudes and skills. These different technologies can facilitate assessment by and for learners. At the same time, they demand that both educators and learners develop deeper and more extensive competences for assessment.

Knowledge cannot be handled like a readymade tool that can be used without studying its nature. Knowing about knowledge should figure as a primary requirement to prepare the mind to confront the constant threat of error and illusion that parasitize the human mind. It is a question of arming minds in the vital combat for lucidity (p. 1) Morin, 1999. Knowing what you know, as Morin suggests, requires an understanding of the cultural, intellectual and cognitive properties and processes that shape knowledge, and the extent to which everything we know is subject to error, illusion and emergence of novelty. Assessment competences require an awareness of these vulnerabilities, and strategies to ensure greater capacity to define knowledge as it emerges from learning processes. The following elements are proposed as a framework for the development of assessment competences, with the hope of becoming better able to grasp emergence and to guard, as much as possible, against error and illusion.

1) Recognising the potential and limits of new and emerging assessment tools

Tools, whether high tech or as simple as a checklist or rubric, are an essential element of assessment. Many new and emerging technologies support greater user engagement in assessment processes, provide the timely

and detailed data needed to diagnose learning needs and connect users to relevant resources, build on effective tracking and monitoring strategies to support persistence toward goals, and scaffold learning challenges. Yet there are also some caveats to keep in mind. The quality of new and emerging ICT-based tools depends largely upon the quality of their design as well as the underlying algorithms. High quality tools will require significant investments in measurement technologies that take into account how people actually learn. Indeed, while cognitive scientists have made a great deal of progress in understanding the processes of learning in different subject domains over the last two decades, progress in measurement technologies has been much slower. Within the learning sciences, there is deeper understanding of how learners: move from novice to expert, develop typical misconceptions, create effective learning environments, and use self-assessment and meta-cognitive monitoring (Bransford et al., 1999; Pellegrino et al., 1999). Emerging technologies, including learning analytics, have ambitions of taking into account these advances in our understanding of learning as a cognitive and social process. However efforts to link technologies to effective

learning and assessment are at the early stages. Ensuring the validity and reliability of assessment tools calls for careful testing and innovative experimentation.

2) Making assessment criteria explicit

While technologies such as blogs, social media and rating services provide opportunities for anyone to express his or her opinion on any number of subjects, the criteria underlying assessments are not always made explicit. Both educators and learners need to be able to clearly identify criteria they are using to make judgments. For example, learners writing blogs need to learn to outline carefully constructed arguments and describe the basis on which they have developed an opinion or made a judgment – a skill emphasized in traditional schooling, but not always applied to the Internet. As consumers of information, learners also need to be able to identify the implicit basis for the evaluator’s assessment when criteria are not clear, and to appraise the validity of their judgments. Are ratings and assessments measuring what they purport to measure? How do others’ assessments relate to the learner’s own values and priorities and can online ratings inform their choices?

3) Strengthening meta-cognition and self-regulation

Meta-cognition (thinking about thinking) and selfregulation (self control) are both central to effective learning and assessment. This includes the way we construct our identities as learners, awareness of effective learning approaches and strategies, and how cognition affects perception and judgment, including the potential for bias and error. Self-regulation places the emphasis on the process of learning rather than the outcome. Assessment may focus on self-monitoring and the development of strategies to support persistence and effort. Here, the kind of tracking programmes mentioned above, that support self-assessment – whether against one’s own prior performances, against clear goals, or against the performance of peers (respectively ipsative-, criterionor norm-referenced assessments) may support the development of effective self-regulation.

Meta-cognition is also vital in understanding the quality of learning, whether toward specific goals or in exploring new ideas. Awareness of the potential for bias and error in assessments, are an important aspect of meta-cognition. Kahneman (2011) notes that, for the most part, we rely upon impressions and feelings and are confident in our intuitive judgments. Such judgments and actions may be appropriate most of the time. However, cognition can also induce people to make errors and biased judgments on a systematic basis. While individuals are capable of  sophisticated thinking, it requires much greater level of mental effort, and the normal human tendency is to revert to intuitive approaches. Awareness of these weaknesses and that they may be particularly dangerous in certain circumstances is a first step toward more effective assessment. More advanced assessment competences mean that individuals are better able to regulate the use of cognitive and intuitive approaches as well as recognize situations when error and bias might be more likely.

4) Judging the quality of information gathered in the assessment process

The quality of information gathered in assessment processes is vital for learning, whether outcomes are known or yet to be discovered. In the case of learning aimed at specific goals or outcomes, assessments of learner attainment are effective only if they are both valid and reliable. Validity means that the assessments measure what they purport to measure, while reliability means the assessment can be repeated and produce consistent results. Validity and reliability are important whether referring to large-scale, high-stakes assessments, classroom questioning, or self- and peer-assessment. The quality of information gathered in assessment is also vital for learning when it involves radical re-framing of ideas to open new ways of thinking and acting and finding novel solutions and tools to address problems. The learning outcome, in these cases, is unknown and assessment is more about testing ideas. It may involve “simple tinkering” or the development of predictive models to systematically test and verify hypotheses. Whether learning outcomes are known or not, both educators and learners need to develop an understanding of and appreciation for the quality of information gathered in different assessment processes. This includes an understanding of how different measurement models mediate that information, and recognition that no single assessment can capture all the information necessary. Multiple measures over time are necessary for more complete pictures. Assessment competences should thus include an understanding of the strengths and weakness of different approaches to measurement, and how they complement each other.

5) Inquiry

Both novice and expert learners may engage in inquiry as a form of deep learning. However, inquiry also requires strong competences to ask meaningful questions and to pursue lines of investigation. Questions, a form of assessment, provide a means to develop understanding, and to identify areas where thinking is unclear. For learners at “higher” levels of expertise, the outcomes of any inquiry may not necessarily be known. Thus, learners, collaborators, educators will need to develop competences to assess learning when information is incomplete and the answers are not clear. They will need to set the goals for different steps of the inquiry process, to develop criteria to gauge the quality of learning, and to develop ways to assess it. Here, the capacity to harness new learning analytical tools to mine “big data” and support inquiry hold particular promise.

6) Sense-making

Individuals, whether working individually or collaboratively, need to make sense of information – to weigh its value and to place it in context. Sensemaking competences include the ability to sort through information when there is too much of it, to recognize when there is too little information, to understand the ways in which individuals and groups shape information based on their own frames of reference, and similarly, the ways in which assessment technologies filter information. Sense making may also require reassessment of beliefs and prior knowledge in order to arrive at new and deeper understandings, and to create new meaning. Indeed, this kind of learning may be considered as “transformative” and can have a profound impact on development (Mezirow, 2011). The willingness to learn and to re-learn, to assess and to re-assess, to experiment and reflect creates the basis for wisdom (Miller et al., 2008).

7) Identifying next steps for learning and progress

When the aim of assessment is to improve future learning, then the information gathered in assessment processes should be linked to next steps for learning. The capacity to diagnose learning needs, and to identify appropriate strategies and resources for further learning, to search for further input from mentors and peers or the broader environment, are thus an essential part of the learning and assessment process. Indeed, this is an essential competence for learning-to-learn.

Assessment has always been an important competence for learning to learn, but it is increasingly central as new socio-economic contexts and new tools have increasingly required and enabled individuals to take charge of their own learning. The above elements represent a first effort at re-defining and deepening assessment competences as part of Morin’s ongoing “combat for lucidity” and knowledge in an emergent world.

Ultimately, assessment competences are about how each individual constructs her or his identity as a learner and as a person. Thus, the standards which guide a learners’ development as well as the ability to assess learning when there are no set standards, the capacity to re-assess and re-frame ideas and prior beliefs, and the ability to define next steps, are all vital steps along the path to lifelong learning.

Learning Analytics: A Platform for Learning

Education has long been a black box where educators and administrators have a limited understanding of the

methods, activities, and investments that produce quality learning. Learners in turn have only a limited awareness of the opportunities offered by tools and technologies to improve self-regulation in the learning process. Over the past decade, three trends have altered information spaces: quantity of data, type of data, and increased computation power. The quantity of data is evident everywhere, presenting challenges for individuals to cope with information overload. New types of data are produced through, and captured by, social media, mobile devices, and, increasingly, automated data capture through sensors (i.e. the Internet of things). Finally, as Moore’s Law has held reasonably firm, computation power has continued to grow exponentially.

As a result of these three trends, business and organizations increasingly recognize that data is an asset – a resource to be managed and leveraged. “Big data” has been coined to describe the growing quantity, and analytic opportunity, of data. PW Anderson, in 1972, prophetically anticipated the challenge facing organizations today: more is different. As data quantity increases, new approaches for mining and drawing insight from that data are needed. Businesses have responded to the new data reality through development of business intelligence. Governments and many organizations (such as OECD) have responded by opening up their data for researchers to analyze. In the education market, interest in learning analytics is growing as well. Learning analytics is the use of intelligent data, learner-produced data, and analysis models to discover patterns and connections within that data, and to predict and advise on learning.

At a basic level, learning analytics relies on some of the concepts employed in web analysis, such as Google Analytics, as well as those involved in data mining. These analytic approaches focus on learner activity through mining clicks, attention/focus “heat maps”, social network analysis, and recommender systems. Learning analytics extends these basic analytics models by focusing on curriculum mapping, personalization and adaptation, prediction, intervention, and competency determination.

Learners constantly put-off data – sometimes explicitly in the form of a tweet, Facebook update, logging into a learning management system, or creating a blogpost. At other times, unintentionally while in the course of daily affairs (or data that is provided by someone else – such as being tagged in Facebook). These data trails are captured by organizations, waiting for some type of analysis that generates insight of value. Amazon, Google, and Facebook use this data to personalize the experience for site users and to recommend additional resources or social connections.

Each new source of data amplifies the potential value of analytics. Most individuals have a fragmented digital profile with bits and pieces of data captured in LinkedIn, Facebook, Google, Twitter, blogs, and closed organizational systems. When these data sources are brought together, especially with physical world data captured by sensors or mobile devices, they can provide organizations with a profile of an individual (the integration of data sources was the primary intent behind the Information Awareness Office’s concept of Total Information Awareness following 9/11 in the USA). The recent development of semantic linked data holds promise in education. For example, “intelligent” data3 (machine-readable semantic or linked data) can 

be combined with learner-produced data, profile information, and curricular data for advanced analysis.

The data trails and profile, in relation to existing curricula, can be analyzed and then used to identify learners who are at risk of dropout or failure. This identification can then be used for automated or human intervention, personalization, and adaptation . Curriculum in schools and higher education is generally pre-planned. Designers create course content, interaction, and support resources well before any learner arrives for a course. This can be described as “efficient learner hypothesis”– the assertion that learners are at roughly

the same stage when they start a course and that they progress at roughly the same pace. Educators generally recognize that this is not they case. Learners enter courses with dramatically varied knowledge levels and progress through content at different paces. As a result, the process used by educational institutions in designing learning is in urgent need of restructuring.

Learning content, for instance, could be computed – a real-time rendering of learning resources and social suggestions based on the profile of a learner, her conceptual understanding of a subject, and her previous experience. Competence (as measured by a degree or certificate) need not even be explicitly pursued. For example, an integrated learning system could track physical and online interactions, analyze how learners are developing skills and competencies, and then compare formal and informal learning with discipline or field of knowledge. This comparison may be possible if a discipline has utilized intelligent/semantic/linked data to define its knowledge goals. Then, the learning system could inform a learner that (you are 64% of the way to a achieving a PhD in psychology, 92% to achieving a masters in science, 100% to achieving a certificate in online learning” and so on). If the learner then decided to pursue a PhD in psychology, the learning system could offer a personalized path forward that adapts constantly to knowledge the learner acquires in the course of work, formal learning, parenting, or the general process of “living life”.

Analytics for pre-defined goals can be approached from a bottom-up or top-down manner. Bottom up involves individual educators using single functionality tools to gain insight into the activities of learners. Fairly simple tools exist for conducting social network or discourse analysis. Additionally, statistics from a learning management system can provide data on login and posting frequency, indicators of learner specific habits. Top-down analytics, in contrast, require the formation of a strategy and vision on the part of an institution. When viewed from an institutional level, analytics can provide valuable information about learner success and failure by leveraging large datasets to reveal interesting correlations. Systems level analytics also gives an organization the ability to plan for automated interventions and develop learner profiles across various data sets. The analytics process, systemically deployed, consists of the stages.

When learning analytics are deployed systemically, several important, and integrated, functionality areas are needed . The analytics engine or module includes the specific analysis being performed on learner data such as social network analysis, predictions of learner failure, and evaluation of the impact of taking different course paths on learner success. The analytics engine draws data from learning management systems and personal learning environments, the social web, organizational learner profiles, and physical world data such as library usage.

The intervention engine is activated when learners exhibit signs of risk for dropout or failure or sub-optimal learning paths and activities. The adaptation and personalization engine uses intelligent learning content and compares learner ongoing activity with the goals or targets of a particular course or learning experience. Finally, dashboards enable end users (learners, educators, and administrators) to query data and visualize how different learners are performing. Many segments of the information industry – music, movies, news – have been dramatically altered by digital information and new options for end users to manage and control information. The financial and accountability pressures facing education, coupled with disruptive advances in technology, present a crisis point for the industry.

Education is adapting and evolving with the changes in today’s technology and information climate. Analytics hold the prospect of enabling changes and restructuring to the processes of teaching, learning, and administration. More importantly, analytics provide a feedback loop to track the impact of change initiatives, in education systems with predefined goals and potentially, as “assessment competences” improve, with open self-referential learning. Today’s recognition of societal complexity and the transformations occurring in the education market are providing important incentives for efforts to construct new platforms for improving the

quality of information that learners and educators use. Learning analytics open up the prospect of providing new resources for leaders, educators, and learners to make a leap in learning productivity happen.

MOOCS - SO MUCH POTENTIAL .....

MOOCs, Massive Open Online Courses, if true to their name, are defined by three elements. Their openness means that they are available to anyone who wants to use them to learn. This logically implies that they are free, removing any financial barrier for even the poorest student. Being online means they are available on the internet. In providing courses, MOOCs represent a major shift in scale beyond open learning objects. They operate at the level of a whole course (or subject) – they provide a coherent learning sequence, with integrated learning materials and formative assessment, all created and managed by outstanding teachers from the world’s top institutions. If a course is of high quality, free (open) and readily accessible (online), it follows that massive numbers of students will grab the chance to get a first rate education for free. This creates scalability challenges. There is solid understanding of how to tackle the engineering of web sites that gracefully handle huge numbers of users. The much less well understood scalability issue is for the teaching, learning and assessment models. The MOOC approach meshes with the acknowledged importance of social interaction among learners as a MOOC can call upon its large community of learners to play two key roles: supporting learning, via discussions, and assessment, based on peer review.

There is a delightful idealism and altruism in the words of many of those driving the MOOC movement. We all agree that quality education is important. We all  know that there is a huge gap between the educational opportunities of the most privileged and the most disadvantaged learners. MOOCS are presented as a means to help close this gap.We can conjure up images of students from the developing world, and the most disadvantaged groups in the first world, as well as lifelong learners with changing learning needs, all slaking their thirst for knowledge, by learning at the feet of the intellectual  giants of the world’s leading research institutions. This is an example of Friedman’s flat world . The wide availability of inexpensive networked computers makes it possible to cater for a large unmet need. Beyond the excitement of the learning opportunities of the actual MOOC courses, a different dimension of promise is in MOOCs as open platforms, built by a new and energetic open source community. Perhaps this will be a revolution in software for authoring and delivering high quality learning opportunities.

Emerging MOOCs have the potential to improve, by exploiting diverse results and techniques from AIE( artificial intelligence in education). MOOC platforms also create new opportunities for new AIE research. They are in particularly interesting for computer scientists working in fields such as Educational Data Mining  and Learning Analytics . Not only can learning-related data from MOOC courses be truly “big” (provided the fallout rate is suitably managed), the open nature of MOOCs seems likely to provide a very heterogeneous student body signing up. These students may interact in ways that are not further structured by established social contracts and roles, making MOOCs an ideal vista for applying Social Network Analysis methods in particular.

Methods from educational data mining and learning analytics can in general be applied for knowledge creation (learning more about learning and interaction, and relevant technologies). They can also serve applied purposes: supporting students, teachers, educational institutions and systems. A rather obvious applied challenge, in light of the mentioned attrition rate, is the automatic identification of students at risk of failing. Similar techniques can be used to “nudge” students who need it, as well as for course- or cohort-based monitoring . We can expect the growth of large collections of learning data, similar to the PSLC Datashop. This can provide a new scale in test-beds for EDM researchers.We can then expect to see more innovative uses of learning data to improve teaching as in the elegant system to generate hints  for students, by drawing on historic data from the paths taken by successful and unsuccessful students.

Pedagogic interface agents are one of the current hot topics in AIE. These anthropomorphic conversational characters have been shown to give real benefits for learning . While one might expect this effect to be short-lived, being of limited value once the novelty has worn off, recent results indicate that interface agents may actually help people stay the course over the long term . They seem to offer promise of a valuable role in MOOCs.

In addition to the general opportunities for research on how to support (on-line) learning with technical means, MOOCs might provide a particular fruitful arena for research on e-portfolio systems, competence management (including assessment), and technical support for lifelong learning (including open learner models). The quality, timing and form of feedback is critical to effective learning. MOOCs currently rely heavily on selfand peer review. These forms already have a recognised place in higher education . However, they are more effective if students are explicitly taught how to do it, a valuable role for AIE systems. Another key form of valuable feedback can be provided for learning contexts for high quality assessment can be automated. There are many systems already for this in domains like programming, mathematics and physics. And AIE research has produced many systems that have been able to give high quality feedback in these classes of well-defined learning domains such as mathematics, physics, and computer programming.

These classes of MOOCs can also be part of a hybrid model. For example, many developing countries have

a large unmet need for skilled IT professionals, where the learning need involved well-defined technical skills.

The most recent MOOCs already have several attractive offerings in this space. This creates the opportunity for employers to create a a learning environment where the MOOC delivers content and basic formative assessment. The employer can complement this by nurturing learning communities. They can conduct summative assessment that determines employment options, a significant motivator for students. the motivator of summative assessment conducted by the employer.

More recently, AIE has moved to ill-structured domains . Notable among these are lifelong generic, particularly the meta-cognitive skills that are a key to success in MOOCs. AIE has demonstrated success in explicit teaching of these skills. The rhetoric about MOOCs refers to personalised learning, with reference to Bloom’s classic 2-Sigma paper about one-to-one tutoring . However, current MOOCs come nowhere near trying to achieve that level of personalisation. One key to the success of AIE systems is in the nature of the personalisation, which is based on a learner model. Indeed, some have argued that very core of AIE is the role of learner model . This core notion of creating an explicit learner model could be readily integrated into MOOCs. Open learner models have been demonstrated to improve learning and they could be a fundamental means for learners to monitor their progress and plan their learning.

It is hard to conceive of MOOCs as having any lasting impact on (higher) education without concern for how

the single MOOC event (course) gets integrated into individual career planning and personal development as

well as into an comprehensive certification framework . Hence, research on how to support the integration of learning events on the individual as well as the societal level will be crucial. The excitment around MOOCs is justified, both in terms of the potential value they offer and the quality of the players who have launched them. What a great opportunity to integrate the lessons, techniques, methods and tools of AIE!

WHAT IS ALGEBRAIC THINKING?

The goal of “algebra for all” has been in place in this country for more than a decade, driven by the need for quantitatively literate citizens and a recognition that algebra is a gatekeeper to more advanced mathematics and opportunities (Silver, 1997; Dudley, 1997). To accomplish this goal, many states, including California, have established algebra as its grade level course for eighth graders (California Board of Education, 1997) . Unfortunately, the data clearly show that all students are not succeeding in algebra in the eighth grade. For example, in 2006 only 22% of California’s eighth graders demonstrated proficiency in a course equivalent to algebra or higher (Kriegler & Lee, 2007). The implication is clear: elementary and middle school mathematics instruction must focus greater attention on preparing all students for challenging middle and high school mathematics programs that include algebra (Chambers, 1994; Silver, 1997). Thus, “algebraic thinking” has become a catch-all phrase for the mathematics teaching and learning that will prepare students with the critical thinking skills needed to fully participate in our democratic society and for successful experiences in algebra.

In this article, algebraic thinking is organized into two major components: the development of mathematical thinking tools and the study of fundamental algebraic ideas. Mathematical thinking tools are analytical habits of mind. They are organized around three topics: problem- solving skills, representation skills, and quantitative reasoning skills. Fundamental algebraic ideas represent the content domain in which mathematical thinking tools develop. They are explored through three lenses: algebra as generalized arithmetic, algebra as a language, and algebra as a tool for functions and mathematical modeling. 

Within the algebraic thinking framework outlined here, it is easy to understand why lively discussions occur within the mathematics community regarding what mathematics should be taught and how. Those who argue that the study of mathematics is important because it helps to develop logical processes may consider mathematical thinking tools as the more critical component of mathematics instruction. On the other hand, those who express concern about the lack of content and rigor within the discipline itself may be focusing greater emphasis on the algebraic ideas themselves. In reality, both are important. One can hardly imagine thinking logically (mathematical thinking tools) with nothing to think about (algebraic ideas). On the other hand, algebra skills that are not understood or connected in logical ways by the learner remain “factoids” of information that are unlikely to increase true mathematical competence.

Mathematical thinking tools are organized here into three general categories: problem-solving skills, representation skills, and reasoning skills. These thinking tools are essential in many subject areas, including mathematics; and quantitatively literate citizens utilize them on a regular basis in the workplace and as part of daily living.

Problem-solving requires having the mathematical tools to figure out what to do when one does not know what to do. Students who have a toolkit of problem-solving strategies (e.g., guess and check, make a list, work backwards, use a model, solve a simpler problem, etc.) are better able to get started on a problem, attack the problem, and figure out what to do with it. Furthermore, since the real world does not include an answer key, exploring mathematical problems using multiple approaches or devising mathematical problems that have multiple solutions gives students opportunities to develop good problem-solving skills and experience the utility of mathematics.

The ability to make connections among multiple representations of mathematical information gives students quantitative communication tools. Mathematical relationships can be displayed in many forms including visually (i.e. diagrams, pictures, or graphs), numerically (i.e. tables, lists, with computations), symbolically, and verbally1. Often a good mathematical explanation includes several of these representations because each one contributes to the understanding of the ideas presented. The ability to create, interpret, and translate among representations gives students powerful tools for mathematical thinking.

The quantitative reasoning is fundamental to success in mathematics, and algebraic thinking helps develop quantitative reasoning within an algebraic framework (Kieran and Chalouh, 1993). Since applications of mathematics rarely require making calculations on “naked numbers,” analyzing problems to extract and quantify relevant information is an essential reasoning skill. Inductive reasoning involves examining particular cases, identifying patterns and relationships among those cases, and extending the patterns and relationships. Deductive reasoning involves drawing conclusions by examining a problem’s structure. Quantitatively literate citizens routinely utilize these types of reasoning on a regular basis.

The line between the study of informal algebraic ideas and formal algebra is often blurred, and the algebra ideas identified here are intended to be studied in concrete or familiar contexts so that students will develop a strong conceptual base for later abstract study of mathematics. In this framework, algebraic ideas are viewed in three ways: algebra as generalized arithmetic, algebra as a language, and algebra as a tool for functions and mathematical modeling.

Algebra is sometimes referred to as generalized arithmetic; therefore, it is essential that instruction give students opportunities to make sense of general procedures performed on numbers and quantities (Battista and Van Auken Borrow, 1998; Vance, 1998). According to Battista, thinking about numerical procedures should begin in the elementary grades and continue until students can eventually express and reflect on procedures using algebraic symbol manipulation (1998). By routinely encouraging conceptual approaches when studying arithmetic procedures, students will develop a network of mathematical structures to draw upon when they begin their study of formal algebra. Here are three examples:

• Elementary school children typically learn to multiply whole numbers using the “U.S. Standard Algorithm.” This procedure is efficient, but the algorithm easily obscures important mathematical connections, such as the role of the distributive property in multiplication or how area and multiplication are connected. These require attention as well.

• The “means-extremes” procedure for solving proportions provides middle school students with an easy-to-learn rule, but does little to help them understand the role of the multiplication property of equality in solving equations or develop sense-making notions about proportionality. These ideas are essential to the study of algebra, and attention to their conceptual development will ease the transition to a more formal study of the subject.

• The widely accepted distance from the earth to the sun is estimated at 93 million miles, but establishing some referants for the meaning of the magnitude of 93,000,000 requires manipulation of ratios and rates and a well-developed generalized number sense.

Algebra is a language (Usiskin, 1997). To comprehend this language, one must understand the concept of a variable and variable expressions, and the meanings of solutions. It involves appropriate use of the properties of the number system. It requires the ability to read, write, and manipulate both numbers and symbolic representations in formulas, expressions, equations, and inequalities. In short, being fluent in the language of algebra requires understanding the meaning of its vocabulary (i.e. symbols and variables) and flexibility to use its grammar rules (i.e. mathematical properties and conventions). Historically, beginning algebra courses have emphasized this view of algebra. Here are two examples:

• How to interpret symbols or numbers that are written next to each other can be problematic for students. In our number system, the symbol “149” means “one hundred forty-nine.” However, in the language of algebra, the expression “14x” means “multiply fourteen by ‘x.’” Furthermore, x14 = 14x, but “14x” is the preferred expression because, by convention, we write the numeral or “coefficient” first.

• The variables used in algebra take on different meanings, depending on context. For example, in the equation 3+x = 7, “x” is an unknown, and “4” is the solution to the equation. But in the statement A(x+y) = Ax+Ay, the “x” is being used to generalize a pattern.

Finally, algebra can be viewed as a tool for functions and mathematical modeling. Through this lens, algebraic thinking shows students the real-life uses and relevance of algebra (Herbert and Brown, 1997). Seeking, expressing, and generalizing patterns and rules in real world contexts; representing mathematical ideas using equations, tables, and graphs; working with input and output patterns; and developing coordinate graphing techniques are mathematical activities that build algebra-related skills. Functions and mathematical modeling represent contexts for the application of these algebraic ideas.

ZERO - THE LENS

If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things. From counting to calculating, from estimating the odds to knowing exactly when the tides in our affairs will crest, the shining tools of mathematics let us follow the tacking course everything takes through everything else - and all of their parts swing on the smallest of pivots, zero. With these mental devices we make visible the hidden laws controlling the objects around us in their cycles and swerves.

Even the mind itself is mirrored in mathematics, its endless reflections now confusing, now clarifying insight. Zero's path through time and thought has been as full of intrigue, disguise and mistaken identity as were the careers of the travellers who first brought it to the West. In the history book you will see it appear in Sumerian days almost as an afterthought, then in the coming centuries casually alter and almost as casually disappear, to rise again transformed. Its power will seem divine to some, diabolic to others. It will just tease and flit away from the Greeks, live at careless ease in India, suffer our Western crises of identity and emerge this side of Newton with all the subtlety and complexity of our times.

My approach to these adventures will in part be that of a naturalist, collecting the wonderful variety of forms zero takes on - not only as a number but as a metaphor of despair or delight; as a nothing that is an actual something; as the progenitor of us all and as the riddle of riddles. But we, who are more than magpies, feather our nests with bits of time. I will therefore join the naturalist to the historian at the outset, to relish the stories of those who juggled with gigantic numbers as if they were tennis balls; of people who saw their lives hanging on the thread of a calculation; of events sweeping inexorably from East to West and bearing zero along with them - and the way those events were deflected by powerful personalities, such as a brilliant Italian called Blockhead or eccentrics like the Scotsman who fancied himself a warlock.

As we follow the meanderings of zero's symbols and meanings we'll see along with it the making and doing of mathematics — by humans, for humans. No god gave it to us. Its muse speaks only to those who ardently pursue her. And what is that pursuit? A mixture of tinkering and inspiration; an idea that someone strikes on, which then might lie dormant for centuries, only to sprout all at once, here and there, in changed climates of thought; an on-going conversation between guessing and justifying, between imagination and logic. Why should zero, that O without a figure, as Shakespeare called it, play so crucial a role in shaping the gigantic fabric of expressions which is mathematics? Why do most mathematicians give it pride of place in any list of the most important numbers? How could anyone have claimed that since 0x0 = 0, therefore numbers are real? We will see the answers develop as zero evolves.

And as we watch this maturing of zero and mathematics together, deeper motions in our minds will come into focus. Our curious need, for example, to give names to what we create and then to wonder whether creatures exist apart from their names. Our equally compelling, opposite need to distance ourselves ever further from individuals and instances, lunging always toward generalities and abbreviating the singularity of

things to an Escher array, an orchard seen from the air rather than this gnarled tree and that.

Below these currents of thought we will glimpse in successive chapters the yet deeper, slower swells that bear us now toward looking at the world, now toward looking beyond it. The disquieting question of whether zero is out there or a fiction will call up the perennial puzzle of whether we invent or discover the way of things, hence the yet deeper issue of where we are in the hierarchy. Are we creatures or creators, less than - or only a little less than - the angels in our power to appraise?

Mathematics is an activity about activity. It hasn't ended has hardly in fact begun, although the polish of its works might give them the look of monuments, and a history of zero mark it as complete. But zero stands not for the closing of a ring: it is rather a gateway. One of the most visionary mathematicians of our time, Alexander Grothendieck, whose results have changed our very way of looking at mathematics, worked for years on his magnum opus, revising, extending - and with it the preface and overview, his Chapter Zero. But neither now will ever be finished. Always beckoning, approached but never achieved: perhaps this comes closest to the nature of zero.

Beauty in mathematics

In this article I will discuss beauty in mathematics and I will present a case for why I consider beauty to be arguably the most important feature of mathematics. However, I will first make some general comments about mathematics that are relevant to my discussion.

Mathematics essentially comprises an abundance of ideas. Number, triangle and limit are just some examples of the myriad ideas in mathematics. I find from experience in teaching mathematics and promoting mathematics among the general public that it's a big surprise for many people when they hear that number is an idea that cannot be sensed with our five physical senses. Numbers are indispensable in today's society and appear practically everywhere from football scores to phone numbers to the time of day.

The reason number appears practically everywhere is because a nuinber is actually an idea and not something physical. Many people think that they can physically see the nuinber two when it's written on the blacltboard but this is not so. The number two cannot be physically sensed because it's an idea.

Mathematical ideas like number can only be really 'seen' with the 'eyes of the mind' because that is how one 'sees' ideas. Think of a sheet of music which is importailt and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a blackboard are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent. 

The nuinber 2 on the blackboard is purely a symbol to represent the idea we call two. Many people claim they do not see mathematics in the physical world and this is because they are looking with the wrong eyes. These people are not looking with the eyes of their mind. For example if you look at a car with your physical eyes you do not really see mathematics, but if you look with the eyes of your mind you may see an abundance of mathematical ideas that are crucial for the design and operation of the car.

So what is this idea we call two? If one looks at the history of number one sees that the powerful idea of number did not come about overnight. As with most potent mathematical ideas, its creation involved much imagination and creativity and it took a long time for the idea to evolve into something close to its current state around 2500 BC. Here is one way to think of what the number two is: Think of all pairs of objects that exist; they all have something in common and this common thing is the idea we call two. One can think of any positive whole number in a similar way. Note that this idea of two is different from two sheep, two cars etc. 

The seemingly simple statement that

20+31=51

is actually an abstract statement, since it deals with ideas rather than concrete objects, and solves infinitely many problems (since you can pick any object you want to count) in one go. This illustrates the incredible practical power of abstraction and many people do not realise that they use abstraction all the time, e.g. when adding. Note that it's not physically possible to solve infinitely many different problems and yet, Hey Presto! it can be done in the abstract in one go. It borders on magic that it can be done.

Abstraction essentially means that we work with ideas and also try to deal with many seemingly different problems/situations in one go, in the abstract, by discarding superfluous information and retaining the important common features, which will be ideas. Many people tend to think of abstraction as the antithesis of practicality but as the above example of addition shows, abstraction can be the most powerful way to solve practical problems because it essentially means you try to solve many seemingly different problems in one go, in the abstract, as opposed to solving all the different problems separately. The latter approach of solving the different problems separately is what people did as relatively recently as less than five thousand years ago by using different physical tokens for counting different objects. For example, they used circular tokens for count- ing sheep and cylindrical tokens for counting jars of oil etc. 

Nowadays, of course, thanks to ab- straction, we just do it in one go as 20+31=51 and it doesn't matter whether we are counting sheep or jars of oil. Clearly, there are much more advanced examples of abstraction but the 20+3 1=5 1 example captures the essential feature of abstraction. These surprises (that number is an idea and addition is an example of abstraction) can actually be very positive experiences for some people and these surprises don't confuse them; in fact it can change their perception of mathematics for the better and make them more comfortable with other more complicated ideas because they are now already comfortable with one abstract mathematical idea, i.e. number. These surprises also enhance the understanding, awareness and appreciation of mathematics for many people. Some people also find it fascinating to know that the idea of number was not always known to humans and was actually created by somebody around 2500 BC. As I said above, before 2500 BC the idea of number had not been created and people used different physical tokens to count different objects.

Now, lok at this pleasing football score:

Louth 1-9 v 1-7 Cork in 1957

Sometimes I use this result, and other examples, to illustrate how number is an idea and why it is so prevalent in today's society. I comment on how the same symbol 9 is used in two different places to indicate two different things. One refers to 9 very satisfying points scored by Louth, while the other refers to 9 hundreds of years. The reason for this is that 9 is just a symbol to represent an idea and that idea can slot into infinitely many different situations. This is one reason why mathematical ideas and abstraction are so powerful and ubiquitous in society today.

The beauty in mathematics typically lies in the beauty of ideas because, as already discussed, mathematics consists of an abundance of ideas. Our notion of beauty usually relates to our five senses, like a beautiful vision or a beautiful sound etc. The notion of beauty in relation to our five senses clearly plays a very important and fundamental role in our society. However, I believe that ideas (which may be unrelated to our five senses) may also have beauty and this is where you will typically find the beauty in mathematics. Thus, in order to experience beauty in mathematics, you typically need to look, not with your physical eyes, but with the 'eyes of your mind' because that is how you 'see' ideas.

From my experience in the teaching of mathematics and the promotion of mathematics among the general public, I have found that the concept of beauty in mathematics shocks many people. However, after a quick example  and a little chat the very same people have changed their perception of mathematics for the better and agree that beauty is a feature of mathematics. One of the reasons why many people are shocked when I

mention beauty in mathematics is because they expect the usual notion of beauty in relation to our five senses but as I said above the beauty in mathematics typically cannot be sensed with our five senses.

Around 2,500 years ago the Classical Greeks reckoned there were three ingredients in beauty and these were:

lucidity, simplicity and restraint.

Note that simplicity above typically means simplicity in hindsight because it may not be easy to come up with the idea initially. On the contrary, it may require much creativity and imagination to come up with the idea initially. These three ingredients above might not necessarily give a complete recipe for beauty for everybody, or maybe a recipe for beauty doesn't even exist. However, it can be interesting to have these ingredients in the back of your mind when you encounter beauty in mathematics. Also, for the Classical Greeks, the three ingredients applied to beauty, not just in mathematics, but in many of their interests like literature, art, sculpture, music, architecture etc.

An example of beauty in mathematics

Example 1. Big sum for a little boy

Here is a simple example of what I consider to be beauty in mathematics. A German boy, Karl Friedrich Gauss (1777-1855), was in his first arithmetic class in the late 18th century and the teacher had to leave for about 15 minutes. The teacher asked the pupils to add up all the numbers from 1 to 100 assuming that would keep them busy while he was gone. Gauss put up his hand before the teacher left the room. Gauss had the answer and his solution exhibits both beauty and practical power. Gauss observed that:

1+100=101,

2+99=101,

3+98=101,

. . .

50+51=101

and so the sum of all the numbers from 1 to 100 is 50 times 101 which is 5050. Notice how Gauss' solution exploits the symmetry in the problem and flows very smoothly. Compare it to the direct brute force approach of 1+2+3+4 .... which is very cumbersome and would take a long time. Both approaches will give the same answer but Gauss' solution is elegant and the other is tedious.

Gauss' approach is also much more powerful than the 1+2+3 ... approach because his idea can be generalised to solve more complicated problems, but you cannot really do much more with the 1+2+3 . approach. This power of the beauty in mathematics happens frequently. For those people who are shocked by the notion of beauty in mathematics, this example from Gauss usually changes their perception of mathematics very quickly for the better and they then agree that beauty can be a feature of mathematics.

Some beautiful visions and sounds can be a consequence of beauty in mathematics. For example, a physically beautiful piece of architecture may be based on the famous number called the Golden Ratio or a beautiful piece of Bach's music may be underpinned by the Fibonacci numbers. Also, certain aesthetically pleasing symmetries in mathematics may produce visually beautiful pieces of art. There are many other examples where beauty, related to our five physical senses, can be a consequence of beauty in mathematics.

Thanks! It's a year now........

I want to thank you for your patronage. I love seeing you and your friends on a regular basis and guiding them to become readers, lifelong learners and blog lovers!

Over the past year,  I have added many new articles to accommodate different aspects of maths learning, 

e-learning and education as a whole increased interest. I am proud to write in a community where the learning is central to people’s lives. There is always an effort from my side to make these articles more interesting and to be as fair as possible. 

“The hunger and thirst for knowledge, the keen delight in the chase, the good humored willingness to admit that the scent was false, the eager desire to get on with the work, the cheerful resolution to go back and begin again, the broad good sense, the unaffected modesty, the imperturbable temper, the gratitude for any little help that was given - all these will remain in my memory though I cannot paint them for others” Quotes Frederic William Maitland .

This is what ‘MATHAMIT’ looks forward to.‘MATHAMIT’ aims at pruning and grooming minds to bloom into flowering personalities and spread fragrance wherever they go.Things do change fast. We are in the second year of ‘MATHAMIT’. From being ‘mentored’ to becoming a mentor themselves. The fact that many of our readers have attained greater heights in their success has held our heads high. The research articles on various topics will give you a glimpse of the various activities that our minds have indulged in, which in turn has helped them to unveil their talents.

Now, I have come up with the second year of ‘MATHAMIT’ and every year we will choose a theme for the cover page, which is either related to recent issues in the field of science and technology or  mathematics or something that is the need of the hour. I will try to focus upon the need of merging our creative scientific ideas, thoughts and research skills with the business forums, which would thereby strengthen the economic status of our country. Knowing that the development of society as a whole solely lies in the hands of the future generation, it is imperative to train the youngsters for the required skill sets. I hope that we attain success in achieving what we have always aimed for i.e. nurturing and bringing out research from Lab to Life and now we could take a step further in the form of service to humanity.

Last but not the least, we take this opportunity to thank our readers and Google+ community, for their cooperation and guidance in releasing this year long journey of ‘MATHAMIT’. My profound thanks to all the students, teaching and the non-teaching people and the ‘MATHAMIT’ committee for willingly coming forward and helping in making the dream of this blog a reality. 

Wishing you a scintillating and happy reading.

Trends from recent thinking on effective learning

• E-learning should not be a mass of online material for individual access without guidance on how to learn from it effectively.
• Courses involving e-learning need to be planned for, and grounded in an understanding of the roles of teachers and learners, of learning, and of how students learn.
• The role of prior knowledge in learning is critical and must be taken into account in e-learning design. Ongoing formative assessment is part of this.
• The brain is a dynamic organ shaped by experiences. Conceptual links are reorganised through active engagement with information in various contexts.
• Learning is an active process. It is the result of carrying out particular activities in a scaffolded environment where one activity provides the step up to the next level of development.
• Learning needs to be meaningful to learners and they should be supported in developing the skill of relating new material to what is meaningful to them.
• Learners should be enabled to become adaptable and flexible experts in their own current and future learning.
• Learning takes time and effective learning practices enable learners to work with materials from a variety of perspectives while they become fully conversant with it.
• Weaving e-learning into existing teaching and learning practices adds more ways for students to be actively and deeply involved with subject area materials.

Most of the evidence related to e-learning is available in case studies of individual courses. There is more evidence related to university than to polytechnic courses. Case studies cover a wide range of subjects, including both skill-based and conceptual. We have focused on blended learning, where e-learning complements some class-based interaction, since that appears more popular with students and teachers, is easier to introduce, and appears to offer some advantages over fully online learning.

Studies comparing student outcomes for e-learning and conventional courses show comparable results in terms of achievement, with indications that student outcomes can be broader if elearning is used well. Student retention shows mixed patterns, and is dependent on a number of factors. Students value the flexibility of e-learning, but it is different from classroom learning, and can demand more.

The main messages from a survey of the available evidence are consistent with the messages from recent thinking on effective teaching and learning.They are:

1.E-learning can improve understanding and encourage deeper learning, if there is careful course design and choice of technology in relation to learning objectives that aim to encourage deeper learning.

2.It can free up face-to-face teacher:student time for discussion, rather than using it to cover information or provide skill practice, depending on the use made of technology.

3.It can improve and sustain motivation by offering interesting tasks and material. .. Students need formative feedback throughout the course. This requires careful structuring and the development of channels and projects encouraging student-student interaction as well as strategic use of teacher time to provide feedback; online tasks, tests, and quizzes are also useful in giving students a picture of their learning progress.

4.Student-student interaction can also be enhanced through careful structuring, creating additional support for learning, and even a “learning community”. Participation in discussion groups, etc. is supported by linking it to assessment or tasks and measures that “matter”.

5.It is important that students have a clear picture of the learning objectives for the course, and that assessment methods reflect and support the learning objectives.

6.Students need very clear course information, and if accessing the course externally, initial face-to-face sessions are valuable to ensure understanding and skills needed to access the web material, to lay the ground for student-student interaction, particularly if some collaborative work is to be done, and for teacher-student web interaction.

7.While asynchronous formats offer students more flexibility, they may also spend more time on a course using the web.

8.The technology used has to be reliable, simple, and easily accessed by students.

9.E-learning is easier for students who are self-managing, which may mean it is easier for mature students.

10.Barriers to making the most of e-learning can arise from students’ familiarity with classroom based methods and assumptions that this is how learning occurs, and from a greater interest in superficial learning to pass a course, than in increasing understanding.

However, increasingly attention is focusing on the creation of tasks, material, and feedback mechanisms and channels that will increase motivation and hence encourage self management, and on course structures, processes, and requirements that provide some additional frameworks for those who need them.

Proportionality,Computataion and Equality - Prerequisite Knowledge for the Learning of Algebra

Proportionality

Fractions commonly appear in beginning algebra in the form of proportions, which provide wonderful examples of naturally occurring linear functions. Because of this, Post, Behr, and Lesh feel that proportionality has the ability to connect common numerical experiences and patterns, with which students are familiar, to more abstract relationships in algebra. Proportions can also be used to introduce students to algebraic representation and variable manipulation in a way that parallels their knowledge of arithmetic.

In fact, proportions are useful in a multitude of algebraic processes, including problem solving, graphing, translating and using tables, along with other modes of algebraic representation. Due to its vast utility, Post et al. (1988) consider proportionality to be an important contributor to students’ development of pre-algebraic understanding. Similarly, Readiness Indicator number 4 focuses on the importance of ratios, rates and proportions in the  study of algebra (Bottoms, 2003).

Proportional reasoning requires a solid understanding of several rational number concepts including order and equivalence, the relationship between a unit and its parts, the meaning and interpretation of ratio, and various division issues (Post et al., 1988). Therefore, these concepts could be considered, along with proportional reasoning, prerequisite knowledge for the learning of algebra.

Computations

In addition to understanding the properties of numbers, algebra students need to understand the rules behind numerical computations, as stated in Readiness Indicator number. Computational errors cause many mistakes for algebra students, especially when simplifying algebraic expressions. Booth (1984) claims elementary algebra students’ difficulties are caused by confusion surrounding computational ideas, including inverse operations, associativity, commutativity, distributivity, and the order of operations convention. These misconstrued ideas are among basic number rules essential for algebraic manipulation and equation solving (Watson, 1990). The misuse of the order of operations also surfaced within an example of an error made by collegiate algebra students that Pinchback (1991) categorized as result of lack of prerequisite knowledge. Other errors deemed prerequisite occurred while adding expressions with radical terms and within the structure of long division (while dividing a polynomial by a binomial) (Pinchback, 1991).

Mentioned by Rotman (1991) as a prerequisite arithmetic skill, the order of operations is also included in Readiness Indicator number 10 (Bottoms, 2003). In fact, this convention has been found to be commonly misunderstood among algebra students in junior high, high school, and even college (Kieran, 1979, 1988; Pinchback, 1991). The order of operations relies on bracket usage; however, algebra requires students to have a more flexible understanding of brackets than in arithmetic. Therefore, according to Linchevski (1995), prealgebra should be used as a time to expand students’ conceptions of brackets.

Kieran (1979) investigated reasons accounting for the common misconception of the order of operations and alarmingly concluded that students’ issues stem from a much deeper problem than forgetting or not learning the material properly in class. The junior high school students, with which Kieran worked, did not see a need for the rules presented within the order of operations. Kieran argues that students must develop an intuitive need for bracket application within the order of operations, before they can learn the surrounding rules. This could be accomplished by having students work with arithmetic identities, instead of open-ended expressions.

Although teachers see ambiguity in solving an open-ended string of arithmetic operations, such as 2 + 4 x 5, students do not. Students tend to solve expressions based on how the items are listed, in a left-to-right fashion, consistent with their cultural tradition of reading and writing English. Therefore, the rules underlying operation order actually contradict students’ natural way of thinking. However, Kieran suggests that if an equation such as 3 x 5 =15 were replaced by 3 x 3+ 2 =15, students would realize that bracket usage is necessary to keep the equation balanced (Kieran, 1979).

Equality

Kieran’s (1979) theory assumes that students have a solid understanding of equations and the notion of equality. Readiness Indicator number 10 suggests that students are familiar with the properties of equality before entering Algebra I (Bottoms, 2003). However, equality is commonly misunderstood by beginning algebra students (Falkner, Levi, & Carpenter, 1999; Herscovics & Kieran, 1980; Kieran, 1981, 1989). Beginning algebra students tend to see the equal sign as a procedural marking that tells them “to do something,” or as a symbol that separates a problem from its answer, rather than a symbol of equivalence (Behr, Erlwanger, & Nichols, 1976, 1980). Even college calculus students have misconceptions about the true meaning of the equal sign (Clement, Narode, & Rosnick, 1981).

Kieran (1981) reviewed research addressing how students interpret the equal sign and uncovered that students, at all levels of education, lack awareness of its equivalence role. Students in high school and college tend to be more accepting of the equal sign’s symbolism for equivalence, however they still described the sign in terms of an operator symbol, with an operation on the left side and a result on the right. Carpenter, Levi, and Farnsworth (2000) further support Kieran’s conclusions by noting that elementary students believe the number immediately to the right of an equal sign needs be the answer to the calculation on the left hand side. For example, students filled in the number sentence 8 + 4 =  __ +5 with 12 or 17.

According to Carpenter et al. (2000), correct interpretation of the equal sign is essential to the learning of algebra, because algebraic reasoning is based on students’ ability to fully understand equality and appropriately use the equal sign for expressing generalizations. For example, the ability to manipulate and solve equations requires students to understand that the two sides of an equation are equivalent expressions and that every equation can be replaced by an equivalent equation (Kieran, 1981). However, Steinberg, Sleeman, and Ktorza (1990) showed that eighth- and ninth-grade algebra students have a weak understanding of equivalent equations.

Kieran (1981) believes that in order to construct meaning while learning algebra, the notion of the equal sign needs to be expanded while working with arithmetic equalities prior to the introduction of algebra. If this notion were built from students’ arithmetic knowledge, the students could acquire an intuitive understanding of the meaning of an equation and gradually transform their understanding into that required for algebra. Similarly, Booth (1986) notes that in arithmetic the equal sign should not be read as “makes”, as in “2 plus 3 makes 5” (Booth, 1986), but instead as “2 plus 3 is equivalent to 5”, addressing set cardinality.

Mathematics a Gatekeeper: A Historical Perspective

Discourse regarding the “gatekeeper” concept in mathematics can be traced back over 2300 years ago to Plato’s (trans. 1996) dialogue, The Republic. In the fictitious dialogue between Socrates and Glaucon regarding education, Plato argued that mathematics was “virtually the first thing everyone has to learn…common to all arts, science, and forms of thought” . Although Plato believed that all students needed to learn arithmetic—”the trivial business of being able to identify one, two, and three” —he reserved advanced mathematics for those that would serve as philosopher guardians2 of the city. He wrote: We shall persuade those who are to perform high functions in the city to undertake calculation, but not as amateurs. They should persist in their studies until they reach the level of pure thought, where they will be able to contemplate the very nature of number. The objects of study ought not to be buying and selling, as if they were preparing to be merchants or brokers. Instead, it should serve the purposes of war and lead the soul away from the world of appearances toward essence and reality. 

Although Plato believed that mathematics was of value for all people in everyday transactions, the study of mathematics that would lead some men from “Hades to the halls of the gods”  should be reserved for those that were “naturally skilled in calculation” ; hence, the birth of mathematics as the privileged discipline or gatekeeper. This view of mathematics as a gatekeeper has persisted through time and manifested itself in early research in the field of mathematics education in the United States. In Stanic’s review of mathematics education of the late 19th and early 20th centuries, he identified the 1890s as establishing “mathematics education as a separate and distinct professional area” , and the 1930s as developing the “crisis” in mathematics education. This crisis—a crisis for mathematics educators—was the projected extinction of mathematics as a required subject in the secondary school curriculum. Drawing on the work of Kliebard , Stanic provided a summary of curriculum interest groups that influenced the position of mathematics in the school curriculum:

 (a) the humanists, who emphasized the traditional disciplines of study found in Western philosophy; 

(b) the developmentalists, who emphasized the “natural” development of the child; 

(c) the social efficiency educators, who emphasized a “scientific” approach that led to the natural development of social stratification; 

and (d) the social meliorists, who emphasized education as a means of working toward social justice.

Stanic noted that mathematics educators, in general, sided with the humanists, claiming: “mathematics should be an important part of the school curriculum” . He also argued that the development of the National Council of Teachers of Mathematics (NCTM) in 1920 was partly in response to the debate that surrounded the position of mathematics within the school curriculum. The founders of the Council wrote: Mathematics courses have been assailed on every hand. So-called educational reformers have tinkered with the courses, and they, not knowing the subject and its values, in many cases have thrown out mathematics altogether or made it entirely elective. …To help remedy the existing situation the National Council of Teachers of Mathematics was organized. 

The question of who should be taught mathematics initially appeared in the debates of the 1920s and centered on “ascertaining who was prepared for the study of algebra” . These debates led to an increase in grouping students according to their presumed mathematics ability. This “ability” grouping often resulted in excluding female students, poor students, and students of color from the opportunity to enroll in advanced mathematics courses . Sixty years after the beginning of the debates, the recognition of this unjust exclusion from advanced mathematics courses spurred the NCTM to publish the Curriculum and Evaluation Standards for School Mathematics (Standards, 1989) that included statements similar to the following:

The social injustices of past schooling practices can no longer be tolerated. Current statistics indicate that those who study advanced mathematics are most often white males. …Creating a just society in which women and various ethnic groups enjoy equal opportunities and equitable treatment is no longer an issue. Mathematics has become a critical filter for employment and full participation in our society. We cannot afford to have the majority of our population mathematically illiterate: Equity has become an economic necessity. 

In the Standards the NCTM contrasted societal needs of the industrial age with those of the information age, concluding that the educational goals of the industrial age no longer met the needs of the information age. They characterized the information age as a dramatic shift in the use of technology which had “changed the nature of the physical, life, and social sciences; business; industry; and government” . The Council contended, “The impact of this technological shift is no longer an intellectual abstraction. It has become an economic reality” . The NCTM believed this shift demanded new societal goals for mathematics education: 

(a) mathematically literate workers, 

(b) lifelong learning,

(c) opportunity for all, 

and (d) an informed electorate.

They argued, “Implicit in these goals is a school system organized to serve as an important resource for all citizens throughout their lives” . These goals required those responsible for mathematics education to strip mathematics from its traditional notions of exclusion and basic computation and develop it into a dynamic form of an inclusive literacy, particularly given that mathematics had become a critical filter for full employment and participation within a democratic society. Countless other education scholars have made similar arguments as they recognize the need for all students to be provided the opportunity to enroll in advanced mathematics courses, arguing that a dynamic mathematics literacy is a gatekeeper for economic access, full citizenship, and higher education. In the paragraphs that follow, I highlight quantitative and qualitative studies that substantiate mathematics as a gatekeeper.

The claims that mathematics is a “critical filter” or gatekeeper to economic access, full citizenship, and higher education.In the today context, mastering mathematics has become more important than ever. Students with a strong grasp of mathematics have an advantage in academics and in the job market. The 8th grade is a critical point in mathematics education. Achievement at that stage clears the way for students to take rigorous high school mathematics and science courses—keys to college entrance and success in the labor force.

Students who take rigorous mathematics and science courses are much more likely to go to college than those who do not. Algebra is the “gateway” to advanced mathematics and science in high school, yet most students do not take it in middle school. Taking rigorous mathematics and science courses in high school appears to be especially important for low-income students. Despite the importance of low-income students taking rigorous mathematics and science courses, these students are less likely to take them. The report, based on statistical analyses, explicitly stated that algebra was the “gateway” or gatekeeper to advanced (i.e., rigorous) mathematics courses and that advanced mathematics provided an advantage in academics and in the job market—the same argument provided by the NCTM and education scholars. The statistical analyses in the report entitled, Do Gatekeeper Courses Expand Educational Options? presented the following findings:

Students who enrolled in algebra as eighth-graders were more likely to reach advanced math courses (e.g., algebra 3, trigonometry, or calculus, etc.) in high school than students who did not enroll in algebra as eighth-graders. Students who enrolled in algebra as eighth-graders, and completed an advanced math course during

high school, were more likely to apply to a fouryear college than those eighth-grade students who did not enroll in algebra as eighth-graders, but who also completed an advanced math course during high school. The summary concluded that not all students who took advanced mathematics courses in high school enrolled in a four-year postsecondary school, although they were more likely to do so—again confirming mathematics as a gatekeeper.

The concept of mathematics as providing the key for passing through the gates to economic access, full citizenship, and higher education is located in the core of Western philosophy. The school mathematics evolved from a discipline in “crisis” into one that would provide the means of “sorting” students. As student enrollment in public schools increased, the opportunity to enroll in advanced mathematics courses (the key) was limited because some students were characterized as “incapable.” Female students, poor students, and students of color were offered a limited access to quality advanced mathematics education. This limited access was a motivating factor behind the Standards, and the subsequent NCTM documents. NCTM and education scholars’ argument that mathematics had and continues to have a gatekeeping status has been confirmed both quantitatively and qualitatively. Given this status, I pose two questions:

(a) Why does our education system not provide all students access to a quality, advanced (mathematics) education that would empower them with economic access and full citizenship? 

and (b) How can we as mathematics educators transform the status quo in the mathematics classroom?

To fully engage in the first question demands a deconstruction of the concepts of democratic public schooling and an analysis of the morals and ethics of capitalism. To provide such a deconstruction and analysis is beyond the scope of this article. Nonetheless, I believe that Bowles’s argument provides a comprehensive, yet condensed response to the question of why our education remains unequal without oversimplifying the complexities of the question. Through a historical analysis of schooling he revealed four components of our education: 

(a) schools evolved not in pursuit of equality, but in response to the developing needs of capitalism (e.g., a skilled and educated work force); 

(b) as the importance of a skilled and educated work force grew within capitalism so did the importance of

maintaining educational inequality in order to reproduce the class structure;

(c) from the 1920s to 1970s the class structure in schools showed no signs of diminishment (the same argument can be made for the 1970s to 2000s); 

and (d) the inequality in education had “its root in the very class structures which it serves to legitimize and reproduce” . 

He concluded by writing: “Inequalities in education are thus seen as part of the web of capitalist society, and likely to persist as long as capitalism survives” . Although Bowles’s statements imply that only the overthrow of capitalism will emancipate education from its inequalities, I believe that developing mathematics classrooms that are empowering to all students might contribute to educational experiences that are more equitable and just. This development may also assist in the deconstruction of capitalism so that it might be reconstructed to be more equitable and just. The following discussion presents three theoretical perspectives that I have identified as empowering students. These perspectives aim to assist in more equitable and just educative experiences for all students: the situated perspective, the culturally relevant perspective, and the critical perspective. I believe these perspectives provide a plausible answer to the second question asked above: How do we as mathematics educators transform the status quo in the mathematics classroom?