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.