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On the emergence of TP-based educational math assistants.
"Recently and largely unnoticed in public, applications in science and technology drove the development of automated and interactive theorem proving (TP) technologies, which have become of major importance for mathematics and computer science. Although based on expressive logical foundations and implemented in a highly trustable way, and although used in some scenarios roughly similar to mathematical tutoring systems, their potential for a wide-spread education technology is unexplored: beyond proving theorems, TPs (1) can manage formal content, check its logical consistency, and verify given problem solutions".
The above text introduces a project proposal, abbreviated "Deducation" (deduction [intersection] education) in 2012, which first time joined experts from Europe's leading TPs, Coq [8] (2) and Isabelle [22] (3). Writing this proposal led to the first collection of publications [28] on the topic "TP Components for Educational Software". On the way of TP towards educational practice there is also a first success: the project "E-Math" (4) will make TP-based educational software available at high-schools following these points:
* Build a web based environment for studying mathematics.
* Provide electronic course material to the web based environment with "structured derivations" method.
* Train teachers in structured derivations as a method of teaching mathematics.
* Carry out pilot studies of "structured derivations" as a teaching method.
The first pilot courses start in autumn 2012 in seventeen schools in four Baltic cities.
Besides first practical success broadly based theoretical work has been done. One line of such work is being pursued at CADGME, the conference on "Computer Algebra and Dynamic Geometry Systems in Mathematics Education" in a specific working group on the topic "TP Components in Educational Software (ConvMath [20], now eduTPS)" (5) (6) (7), so it seems time to give a survey on this topic.
First the question arises, what is this TP and where does it come from. There is an overwhelming palette of educational software, most of it is based on well established standardised components, on Computer Algebra Systems (CAS), Dynamic Geometry Systems (DGS) and Spreadsheets--so, what does existing software not cover?
Furthermore, the theory of mathematics education has been extended with theories covering the introduction of Information and Communication Technologies (ICT) into education. A highly elaborated body of knowledge reflects a wealth of aspects, cognitive, socio-cultural, collaborative and social aspects. Based on these theories "new software have been developed with potential impact on all phases of education and on informal contexts of education"--a citation from p.2 of the 17th ICMI Study on "Mathematics Education and Technology--Rethinking the Terrain" [14]; however, this thorough accountancy of the state-of-the-art does not mention TP.
Above the importance of TP for mathematics and computer science has been mentioned (which is justified by significant contribution of TP within "Formal Methods" supporting engineers to master ever increasing complexity of technical systems)--but this does not imply that TP needs to be introduced to education, in particular not at high-school. So, what do TP's features promise for mathematics education? And if the contribution is really fundamental to mathematics, does it carry over to STEM (8) education? What are the advantages for education and what impact can be expected for the science of STEM education?
The paper will tackle these questions as follows: [section]2 recalls some preliminaries: existing prototypes of TP-based systems in [section]2.1, general adoption of software in STEM education in [section]2.3 and conceptional foundations in "reasoning"--as foundations of both, mathematics and TP, are discussed in [section]2.3. The main section [section]3 investigates TP's features promising for education ([section]3.1), shows that the wide coverage of TP technology allows for "software models of math" ([section]3.2) and is an integrative concept for math software ([section]3.3). Finally some predictions are tried on impact to be expected for the practice and research in STEM education in [section]4.1 and [section]4.2 respectively, as well as on educational administration ([section]4.3), which in [section]5 conclude with the announcement of a new generation of TP-based educational math assistants.
2 Mathematical Software and STEM Education
Starting from some notes on some prototypes of TP-based educational math assistants in [section]2.1 a wide perspective is opened: A perspective on all mathematical software which became relevant for education in [section]2.2, because TP systems are distinct from CAS (9) and other kinds of software, as CAS are distinct from DGS 10--and nevertheless, TP will be considered as integrative basis for both, CAS and DGS. Another perspective recalls the fact that "reasoning" is a conceptual base of both, of TP technology and of mathematical thinking.
2.1 Prototypes of TP-based Educational Math Assistants
Any description of TP-based systems influential in education would require lots of space and would, nevertheless, never be complete. The selection below is rather arbitrary and focuses features typical for TP and relevant for further discussion.
GCLC is a tool for visualizing and teaching geometry [15] (11), and for producing mathematical illustrations. GCLC provides easy-to-use support for many geometrical constructions, isometric transformations, conics, parametric curves, flow control, etc. The basic idea behind GCLC is that constructions are formal procedures, rather than drawings. Thus, in GCLC, producing mathematical illustrations is based on "describing figures" (in a custom geometry construction language) rather than of "drawing figures". This language is integrated with automated TP. So GCLC can prove (by logics, not by numerical computation) that the intersection [O.sub.1] of [bar.CB] and [bar.AC] is the same point as the intersection of [bar.CB] and [bar.AC], see the line at the bottom of the left window in Figure 1 on p.113:
Socos is a programming environment [3, 4] (12) where correctness proofs are developed hand-in-hand with the program. The loop invariants are formulated before the code. Once the invariant structure is defined, the code is added in small increments while continuously being verified to be consistent with the invariants. The programmer draws the invariant diagram in a graphical IDE (implemented as a plug-in to Eclipse) and the tool generates the verification conditions and checks them automatically. Figure 2 shows one (of the infinitely many possible) diagrams, where the nodes, are labeled by predicates (invariants), and the arcs are labeled by statements--while all these formulae and their logical inter-dependencies are checked by use of a TP:
Structured Derivations is the format for interactive calculations [2] checked by TP in the EMath project mentioned in [section]1. The front-end connects to a server via Internet, allowing student solutions to be checked on request. All errors found in the proposed solution will immediately be reported back to the student (and possibly to the teacher). The main challenge here is ensuring that the feedback is timely and presented in a comprehensible format, so that interaction with the system in an explorative way continuously motivates the student to improve his or her reasoning skills. Figure 3 on p.115 shows, how close the format is to traditional work with paper and pencil:
Isac is an experimental system (13) based on the TP Isabelle [22] and aiming at STEM education. The focus of the experiments is "next-step-guidance" [21]: if the learner gets stuck during interactive construction of a problem solution, the system gives various kinds of help at any step: one or several rules to be applied at the current step and/or the formula of the step, probably with gaps to be filled, displays underlying specifications, algorithms or theories on request--the latter is shown in the right window in Figure 4 on p.116:
2.2 Adoption of Mathematical Software in Education
Since software has been used in mathematics education in a serious way, such software has been either adopted directly from engineering domains or has been developed further from products previously established in science and engineering (the major exception, DGS, will be mentioned). So, when looking at the future of educational mathematics software, a survey on mathematically biased software seems appropriate.
The survey in Fig.2.2 on p.117 has three columns: the left one shows three academic disciplines, the middle one shows the domains, software tools have been created for, and the right column shows the products shaped out by standardisation over the years (the spatial distances between the items try to express close/loose relations):
In the left column the discipline of (computer) mathematics is the source of the software-tools in the middle column, the disciplines of applied sciences in academia and industry (introduced as "STEM" on p.112) drove and still drive further development, usually leading to some kind of standardised products listed on the right. Remarkably, development of Computer Algebra (Macsyma 1968) started at the same time as development of Theorem Proving (Automath 1967); however, request from academia and industry drove development to mature CAS very soon, while TP only recently reaches relevance in industry.
Relevant for "math. education" and development of respective software are windows of opportunity at the transition of software from academia to industry: At this transition Spreadsheets and CAS have been adapted to educational needs (DGS are the major exception (14), developed as educational tools from scratch); now this kind of window can be exploited for TP as well, now the respective systems are open source and interested in attracting future scientists and engineers. In a few years the successors of Socos, Coq or Isabelle might have adapted to the needs of industrial practice such that requests from the side of education will be not as well accepted as presently.
"TPS" above could stand for TP-systems; however, it is marked by "?!?" due to the authors opinion, that the primary kinds of software used in high-schools will be TP-based systems where TP technology works in the background as done in the four examples presented in [section]2.1 above and discussed in detail in [section]3.3 below. However, TP-systems (academic systems downgraded for education, probably in several levels, for instance [40]) are expected to provide smooth transitions to academic mathematics in the higher grades of high-school.
2.3 "Reasoning" in TP technology and in Mathematical Thinking
Rigorous reasoning is fundamental to mathematical thinking (such that an approved publication like [14] not even lists it in the Subject Index, apparently due to ubiquitous occurrence).
However, mathematisation (see below) of reasoning succeeded only in the last century due to the work of logicians like Russel and Whitehead, Godel, Church and Tarski. And mechanisation of logics in TP succeeded only in the last decades: The first "proof of the Four-Colour-Problem by Appel and Haken was not accepted as a proof for good reasons: Appel and Haken could not prove that their program did not contain errors in 1976. Now, in 2005, Georges Gonthier used the TP Coq to create a surveyable proof [9]. In 2003 another proof of a difficult problem, Kepler's conjecture, was judged 99% correct by twelve referees after four years of work--so the flyspeck project(15) has been set up to produce a formal proof of the Kepler Conjecture. Since all that is very new, there still goes the saying, that a serious mathematician uses a computer only for LATEX, for mail and for retrieving references.
As rigorous mechanical proving is still controversial in academic mathematics, it is even more in mathematics education. The 19th ICMI Study [18] states "re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification (including dynamic graphic software) has already produced several theoretical frameworks, giving rise to many discussions and even heated debates."
In science of ... SW-tools for ... ... standardized as .. mathematics geometry DS numerical Spreadsheets computation (SSH) applied sciences symbolic compuatation math. education Computer algebra CAS Macsyma 1968 Theory Proving (TP) TPS |?| Automath 1967
This paper circumvents such debates (16) by approaching mathematics education by the way of mathematical software, asking: How can TP technology support development of mathematical thinking? Taking PISA's most recent competence model for mathematics [24], the above question can be approached by a more concrete one: Which capabilities underlying mathematical thinking can be supported by TP in principle? PISA's "seven fundamental capabilities" can be supported by TP technology as follows:
1. Communication: ... perceiving the existence of some challenge and recognizing a problem situation ... is not specific to TP, but general to ICT (with presentation tools, mail, information retrieval).
2. Mathematising: ... transforming a problem defined in the real world to a strictly mathematical form ... is a prerequisite for TPS support such that even libraries of formal specifications need to be prepared; specification is indispensable, but might be hidden from learners not specifically interested.
3. Representation: ... selecting, interpreting and using a variety of representations to capture a situation ... is well implemented in DGS showing geometric and analytic representation of objects if requested; TP technology can add logical representation, ready for investigation in case of particular interest.
4. Reasoning and argument: ... logically rooted thought processes that check a justification that is given ... have a strong counterpart in TP technology where each step of a problem solution must have a mechanical justification; these justifications are human readable and can be augmented by multi-media explanations.
5. Devising strategies for solving problems: ... critical control processes that solve problems ... involve inquiry-based processes which efficiently can be supported by software: "next-step-guidance" (see [section]3.1) allows to quickly try alternative specifications and algorithms and to accurately compare them.
6. Using symbolic, formal and technical language and operations: ... within a mathematical context ... is considered a major obstacle already in using CAS; however, TP-based software has formal specifications and algorithms in the background, which can be used for additional help (last not least by "next-step-guidance").
7. Tools: ... being able to make use of various tools that may assist math activity ... while the comprehensive assistance expected for TP-based tools suggests to rethink, if separation of this point from the other six points is still appropriate.
So, all capabilities relevant for doing mathematics are related to some support by TP-based technology in the enumeration above--but caution is called for methodological reasons: PISAs "seven fundamental capabilities" represent branches of a tree, which is firmly rooted in the grounds of science of mathematics education; the seven branches cover mathematical capabilities in full completeness at the well settled state-of-the-art in the science of mathematics education.
Above, specific kinds of TP support are attached to the tree's branches--to the authors best knowledge, this is the first attempt to give a survey on TP features from the point of view of STEM education. So this assemblage of TP features is necessarily ad hoc (like hanging strange Christmas tree balls into the branches of a well-rooted Christmas tree; and in particular, the conclusion "that TP technology would support everything viewed as essential for math education" is invalid). This might be particularly painful to TP experts, who know their science well rooted in mathematics and who would like to see some of their well-elaborated structures in this assemblage. Structuring this new field will, in the author's opinion, require joint efforts for some years.
However, the author is convinced, that the above ad hoc relations provide starting points useful for mutual approaches between education and TP. And probably, the structure suggested above might serve future field-test on TP-based mathematics assistants in educational practice.
Staying optimistic that way, we see the above points also outlining phases and iterations in problem solving, and conclude: almost the full range of mathematical problem solving can be supported by software. With respect to these possibilities the statement "marginalization of technology [...] points, in part, to a failure to theorize adequately the complexity of supporting learners to develop a fluent and effective relationship with technology in the classroom"'([13] cited from [14]) suggests the comment: mastering such complexity cannot be accomplished by technology limited to (numerical and symbolic) computation (by CAS) and drawing (by DGS)--TP technology adds logic and reasoning as a feature fundamental to mathematical thinking.
The subsequent section will show details of how TP-technology can support a wide range of mathematical problem solving.
3 TP's Features Promising for Education
Combining the two facts, that (1) reasoning is fundamental to mathematical thinking and (2) TP technology implements reasoning and logics, the wide coverage of TP support for doing mathematics does not come as a surprise anymore. The following features are a preliminary account collected from the four TP-based prototypes already described, now restructured alongside educational points of view.
3.1 Technological Features: Flexibility, Transparency, Guidance
The various features of many TP-based systems have been distilled into three groups by discussions in the three CADGME working groups and during proposal writing for Deducation. So the three groups are a collection from existing systems' features, and not some abstract requirements capture. The description of the features will closely refer to the four TP-based prototypes introduced in the above [section]2.1.
Flexibility and reliability in checking user input is the main contribution of TP to educational software: given a logical context from a formal specification, an input formula establishes a proof situation--the formula must be (automatically) derivable from the context. Such automation allows to support stepwise problem solving in software as shown in Figure 3 on p.115 for steps in a calculation; such support is possible for all kinds of steps.
Also DGS begin to include TP methods [19]; for instance, GeoGebra (17) already proceeds from algebraic methods to TP methods [27] which come close to human readable proofs. In order to work, TP requires formal specification of the problem to be solved; this is specifically interesting for geometry, because it involves elementary clarifications, for instance what a triangle is, see Figure 3.1. In order to prove {identical [O.sub.1] [O.sub.2]} from Figure 1 on p.113, TP requires the "non-degeneracy conditions" that the points A, B and C are not collinear and A = B = C. Such specifications are easily comprehensible (18).
And if a student experiences the invariant M by dragging the points A, B, C, she or he might be interested not only in a dynamically linked algebraic representation of the geometric elements, but also in a dynamically linked logical representation shown on request. Questions like "What happens with a proof in a degenerate case?" link geometric intuition with formal mathematics in a novel and worthwhile way.
Transparency of the systems is prepared by TPs like Coq and Isabelle following the "LCF-paradigm": in these TPs all knowledge is mechanically deduced from the basic axioms and definitions of mathematics--following Russell's principle of "honest toil" ([32] p.71). And this knowledge is represented in a format which is not only interpreted mechanically by the TP, but which can also be read by humans.
The reader is invited to look-up some of the knowledge mechanised by Isabelle (19); the theories Groups, Rings, Fields involve only few technicalities, also the definitions of numbers in Int, Rat, RealDef, Complex. So, for instance, a learner facing a term like 2 * n can be mechanically guided to the definition of multiplication, to the proof of commutativity of multiplication (proved for Cauchy Sequences), etc. The issue is not any more to create that knowledge, but to filter off overwhelming details. The upcoming graphical user-interface Isabelle/jEdit [38] already allows to delve into underlying knowledge just by clicking tooltips.
The scope of the knowledge already exceeds the range required for high-school mathematics as well as for STEM education (see MacLaurin and Taylor Series, HOLJMultivariate-Analysis, etc), but the kind of knowledge is not all what is needed for TP-based systems; for instance, given this knowledge one can easily prove (20) . . .
fixes x :: "'a :: {idom, number-ring}"
shows "[x.sup.2] * y = [x.sup.2] & x * [y.sup.2] = [y.sup.2] [left right arrow] x = 1 & y = 1 | x = 0 & y = 0" by algebra
... but one has to find the solution first: the knowledge widely lacks algorithmic parts. Incorporating algorithmic knowledge from CAS into the logically reliable framework of TP is concern of ongoing R&D.
Next-step-guidance is the most notable adaption of TP technology to educational needs. Given the above mentioned feature of checking input allows to automatically generate feedback, when a formula cannot be derived from the context--then a learner might get stuck without knowing how to proceed towards a solution of a problem. In this case the system needs to "know the next step", and this feature can be accomplished by Lucas-Interpretation [21] combining deduction with computation.
The application of Lucas-Interpretation is connected with incorporating algorithmic knowledge into TP as mentioned; the CAS algorithms, programmed in a TP-based programming language [10], become verified and also accessible to step-by-step investigation.
Exploitation of next-step-guidance for adaptive user guidance in a TP-based educational math assistant are under construction [7].
3.2 Interactive, Complete and Transparent Models of Maths
TP's features identified so far relevant for education suggest a new way of thinking about the role of computers, infeasible without TP.
"In the old way of thinking, computers were seen as human tutors and evoked a vision of the teacherless learning environment. New avenues for using technology take advantage of, rather than marginalize, teacher, task, and classroom cultures" [25]. Confirming the teachers' role another vision can be added: the computer as a model of mathematics itself. Now, what are the features of such software models of mathematics?
"Interactive" models are all of the four prototypes to a high degree, such it is possible to learn by just interacting with the system. The 1SAC prototype goes a step further and strives for learning the same way as a good chess-program allows to learn playing chess just by trying some moves, going back and trying another strategy, probably by changing role with the system and watching how the system copes with a certain situation.
Quite analogously, TP's ability to generously and reliably check user-input, allows to model stepwise problem solving and to emulate traditional paper-and-pencil feel. Added next-step-guidance, the learner and the system can cooperate in going step by step towards a solution until the system (automatically) proves that a solution has been found (by proving the so-called post-condition of the problem's specification):
The learner might do some steps, explore several variants or might get stuck and request help; and the system checks the steps giving feedback, and adapts help to the behaviour of the learner by more or less gaps to fill in to the next step, etc.
Next-step-guidance also makes the problem with formatting input vanish: learners have <next>-step button and learn from just observing the system, see Figure 4 on p.116.
"Transparent" models allow to open up "black boxes" on the spot in a calculation and see intermediate steps. All prototypes previously described are clearly designed as "transparent models". Isabelle's feature of proof reconstruction supports such transparency; in case of rewriting, the term rewrite systems need to be modularised such that traces group rewrites into comprehensible "big" steps.
Since in a TP-based system each step has a mechanical justification related to the logical context, the learner can request justification for each step. Starting from the step related knowledge can be investigated; the reader is reminded the respective paragraph in [section]3.1. This feature of maths models goes significantly beyond the features of chess-programs: looking at the software mechanisms would not help learning chess--while look at these mechanisms, requested by learners due to spontaneous interest, definitely has the potential to help: the models advocated here are based on TP and thus on logics.
Also, if an input formula needs several steps for (automated) derivation from the context (which might be hidden in the first go), the learner might investigate the intermediate steps and be surprised, how many rewrites her or his "intuitively justified" step requires on the mechanical level.
"Complete" models cover the whole problem solving process, unlike CAS or DGS, which are a particular tools for certain parts within the whole process (the reader is reminded of the seven fundamental capabilities for mathematical problem solving in PISA's competence model in [section]2.3):
In a TP-based model work on the computer starts with a meaningful problem (see, for instance, the balcony in Figure 4 on p.116); the initial formalisation and specification might be prepared by an author (and even hidden from novices) ; the steps are supported by the "interactive" and "transparent" model mentioned above until the specification's post-condition can be proved by the system (which is most likely hidden from most learners and done automatically behind the scenes).
The envisaged models are expected to be complete also in the sense, that they represent all steps of problem solving on the screen (in several layers, some of which might be hidden); several techniques of zooming and folding/unfolding can support changing from detail to survey. This kind of completeness of representation also makes an issue vanish, raised in [37] by the observation, that students start to develop debatable styles of protocol for their problem solutions using CAS and DGS. In any case, logical representations seem most fruitful when being optional and "dynamically linked" like the analytic representation in DGS.
The above has already been anticipated from an educational point of view: "Feedback in intelligent tutors and CAS allow for the 'playability' of calculus and functions in the way that geometry has become 'playable' through dynamic geometry environments. The calculator can perform the microprocedures and let the student focus on the macroprocedures, which require higher-level processes" ([11] cited by [25])--this vision becomes much more reachable by TP-based technology now.
3.3 Integrative Concepts rather than Separate Systems
Figure 1 on p.113 shows a prove statement, which looks like an particular add-on to DGS. But [section]3.1 showed that proving has formal specification as a prerequisite--consequently such a DGS gets another conceptional foundation based on logics. And if a DGS adopts the conceptual foundations into the technological basis, then the advantages of TP become available like next-step-guidance in stepwise construction and the others.
In DGS the process of integrating TP has already started [27], and there are good reasons to expect continuation of this process.
Formula-based systems, for instance those shown in Figure 3 on p.115 and in Figure 4 on p.116 face the challenge to acquire CAS-functionality. Since CAS are themselves not based on logics, are not reliable (for instance, drop solutions of equations, are inconsistent with "multivalued functions") and are black boxes, meeting this challenge involves great efforts. These efforts cannot be accomplished by development of educational systems alone; rather development of educational systems is expected to take profit from ongoing and planned development aiming at industrial application.
Finally coming back to "TPS!?!" in the figure on p.117 with the considerations just above, it appears straight forward to see TP as unifying conceptual base for a process of system integration already started. For instance GeoGebra already integrated Spreadsheet, CAS and most likely will continue with TP--just following the requirements of educational practice, where teachers are just overwhelmed if forced to use separate tools for solving one problem in one lesson.
4 Expected Impact
The previous section showed the directions TP-based educational software might develop to; it also made clear the considerable amount of efforts to realize these developments. Such efforts need to be justified by clarification, what the use is for these efforts.
With regard to the deep changes in educational software's features, impact is expected to all aspects of education in the full range of STEM (science, technology and engineering), the research on education as well as educational administration; software development in general is affected, too.
4.1 Impact on Practice in STEM Education
"Certain uses of ICT give pupils access to advanced mathematical ideas, which are not currently considered in traditional curricula at the elementary and secondary school education level. These possibilities rely mainly on the potential of the ICT environments to facilitate learners in making crucial transitions towards a mathematical way of thinking. In this way, ICT can significantly alter how didactic and learning trajectories have been traditionally conceived" [33] p.219. Such ideas comprise mathematics of variation, modeling and generalization [17, 30], or infinity and infinite processes [16, 34]--and ideas about justification in different levels of rigor, theorem, axiom, definition, proof, formal system, formal reasoning can be seen in the same line.
Probably curricula are right not to include such ideas in a too concrete manner: students' comprehending fundamental ideas and gaining deep insight can hardly be planned within a certain topic or a single lesson. On the other hand a "good number of studies have shown that computer environments can play a role so that students, of very young ages, may works with sophisticated mathematical ideas" [33] p.218, and "ICT seem to facilitate transitional processes that have previously be reported as being highly complex for the vast majority of students, such as transitions from the particular to the general; from what is concrete to what is abstract; from intuitive perception to formal thinking; from non-mathematics to mathematical representations; etc." [33] p.219
The key in supporting such transitional processes seems to be individualization of learning: "We also recognize that there may be diverse ways or paths for students to construct or develop mathematical thinking and problem solving competencies. We contend that student's use of different digital - and representational - means or technologies, or in accordance, to the facilities or potential associated with each tool" [33] p.219.
This key has been addressed by the repeated confirmations above, that presentation of logical facts may not be isolated from concrete steps in concrete calculations or in the construction of concrete geometric objects--rather logical facts shall come in a "dynamically linked presentation", in the same way optional as is analytic representation linked with geometric objects in DGS.
The above "may not" and "shall" call for intimate cooperation between practice in education, software development and educational research. Already clear seems, that TP-based systems particularly promise to support open learning scenarios, inquiry-based and independent learning.
Impact on settings in informal education and on personalized ubiquitous learning paradigms can hardly be estimated. Just as an example, the TP-based system presented in Figure 3 on p.115 would allow a web-based public competition on solving equations in analogy to the competitions between Italian nobles during Renaissance. Technology-driven developments alongside "cloud computing" with TP-services called from handheld devices [39] and students doing maths homework by that way could well overtake professional expectations like in [12].
4.2 Impact on Research in STEM Education
The novel promises of TP technology, the full coverage of software support for phases in mathematical problem solving and the re-invocation of methodological foundations of mathematics suggest to rethink ontological questions about mathematical objects and processes and learning mathematics. Some of these questions might be:
Papert's Principle [26]: "Some of the most crucial steps in mental growth are based not simply on acquiring new skills, but on acquiring new administrative ways to use what one already knows"--is, for instance, "linked dynamic representation of logical items" an appropriate support for new ways to use knowledge about "justification in different levels of rigor, theorem, etc"?
Can acquisition of such knowledge, apparently belonging to "advanced mathematical thinking" [36], be boiled down to "webbing" and "situated abstraction" [23]? If "the idea of webbing is meant to convey the presence of a structure that learners can draw upon and reconstruct for support - in ways that they choose as appropriate for their struggle to construct meaning for some mathematics" (p.108)--can TP-based educational software "designed in such a way that expressing generality can be made a central component of the computational setting and, thus, that webbing and situated abstraction are co-emergent constituents of the mathematical learning process" (p.122)?
"The driving question of what new types of mathematical knowledge emerge as a result of access to digital technologies"[25] p.136 has been answered without regarding TP technology so far-- so what are the answers when including TP into the selection of base technologies for educational software?
The above questions address long-term impact of early access to powerful mathematical ideas as discussed in [33]. "However, research into ICT-based learning trajectories is still in its infancy. From a theoretical perspective, the idea of learning trajectories in the context of digital technologies still needs to be developed" [33] p.220. [section]3 tried hard to show how automated TP-services can be more or less be hidden in educational software; however, TP implements reasoning on a rigorous formal level--and the question, how to make the reasoning transparent to learners in which levels of learning, is really challenging. Particularly interesting seems the question, how the gap between "intuitive" high-school mathematics and "formal" academic mathematics might be bridged by TP-based software.
A further trajectory crossing over science, technology and engineering (STEM) is established by mathematics as basic thinking technology; also this trajectory is considered urgent (21) but it is not yet in the focus of present research.
Another kind of impact on research can be expected from the ability of TP-based software to produce high-level protocols of user-interaction; high-level means, that they convey interactions with notable semantics. This ability can be implemented at the level of abstraction, where input formulas are checked and accepted or rejected if they cannot be derived from the context. On the same level development of adaptive dialog guidance can exploit next-step-guidance ([section]3.1) which is at the very beginning [7], goes far beyond CA-based systems [35] and calls for involvement of educational research.
4.3 Impact on Administration of Education
Beyond the expectation, that TP-services will be available "from the cloud" on handheld devices for everyone ubiquitously, at least two further points will affect planning and administration of education.
One point is (probably central) availability of data from logging high-level interactions in stepwise problem solving on TP-based systems. For instance, this is already possible in the E-Math project mentioned in [section]1, and it might develop further [39].
Another point is more look-ahead and will be relevant as soon as the coverage of mathematics knowledge mechanised in TP-based systems will go towards 100%: Given that knowledge, mechanised support is feasible to manage cross-institutional courses of study: prerequisites can easily be identified when changing from one institution to the other.
5 Conclusions
TP's features, flexibility, transparency and guidance, enable design of software as interactive, complete and transparent models of mathematics. These advancements in technology are expected to have impact on STEM education, on educational research and on administration. Combining advancements in technology and expected impact suggests to conclude with: Let's announce the emergence of a new generation of TP-based educational mathematics assistants!
The domain of education is wide and multifaceted, and so is the domain of mathematics and respective software. So, if the latter focuses TP technology y, the former focuses STEM education [DELTA] as proposed by the European Commission [29], we can combine both in the following picture:
The "announcement of a new generation" might join efforts in both domains to combine [nabla] and [DELTA] to a common focus [nabla]/[DELTA}. Such joint efforts might entail TP experts' acknowledgment that they have to offer specific services to educational software rather than simply downgrading their systems for education; and it might entail educators' acknowledgment, that mathematical software becomes more flexible and general, more adaptive and appropriate than before, if given logical foundations.
The author owes a great debt of gratitude to the members of the working groups at CADGME for the many discussions helping to identify software requirements in the practice of mathematics education. The colleagues preparing the Deducation proposal helped to clarify facts about what is available in TP already and what can be accomplished in the near future; very special thanks to this group ! Last not least I express my thanks to the referees, who helped a lot to narrow the points of view from education and from TP technology.
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[2] R.-J. Back. Structured derivations: a unified proof style for teaching mathematics. Formal Aspects of Computing, 22(5):629-661, 2010.
[3] Ralph-Johan Back, Johannes Eriksson, and Linda Mannila. Teaching the construction of correct programs using invariant based programming. In Proc. of 3rd South-East European Workshop on Formal Methods (SEEFM07). South-East European Research Centre (SEERC), 2007.
[4] Ralph-Johan Back, Johannes Eriksson, and Magnus Myreen. Testing and verifying invariant based programs in the SOCOS environment. In Proc. Of the International Conference on Tests And Proofs (TAP), volume 4454 of Lecture Notes in Computer Science, pages 61-78. Springer, 2007.
[5] Michael J. Beeson. Design principles of Mathpert: Software to support education in algebra and calculus. In N. Kajler, editor, Computer-Human Interaction in Symbolic Computation, pages 89-115. Springer, 1998.
[6] H Chick, K Stacey, and J. Vincent, editors. The Future of the Teaching and Learning of Algebra. Proceedings of the 12th ICMI Study Conference, Melbourne, Australia, 2001. University of Melbourne.
[7] Gabriella Daroczy and Walther Neuper. Error-patterns within "next-step-guidance" in TP-based educational systems. submitted to EJMT.
[8] Coq development team. Coq 8.3 reference manual. http://coq.inria.fr/reman, 2010. INRIA.
[9] Georges Gonthier. The four colour theorem: Engineering of a formal proof. In Deepak Kapur, editor, ASCM, volume 5081 of Lecture Notes in Computer Science, page 333. Springer, 2007.
[10] Florian Haftmann, Cezary Kaliszyk, and Walther Neuper. CTP-based programming languages ? considerations about an experimental design. ACM Communications in Computer Algebra, 44(1/2):27-41, 2010.
[11] K. Heid. Technology in mathematics education: tapping into visions of the future. In W.J. Masalski, editor, Technology-Supported Mathematics Learning Environments: NCTM 67th Yearbook. NCTM, Reston, 2005.
[12] C. Hoyles, I. Kalas, L.Trouche, L. Hivon, and R. Noss. Connectivity and Virtual Networks for Learning, chapter 22. Volume 13 of Hoyles and Lagrange [14], 2010.
[13] C. Hoyles and R. Noss. What can digital technologies take from and bring to research in mathematics education? In A.J. Bishop, M.A. Clements, C. Keitel, J. Kilpatrick, and F.K.S. Leung, editors, Second International Handbook of Mathematics Education. Kluwer, Dordrecht, 2003.
[14] Celia Hoyles and Jen-Baptist Lagrange, editors. Mathematics Eduation and Technology--Rethinking the Terrain. The 17th ICMI Study, volume 13 of New ICMI Study Series. Springer, 2010.
[15] Predrag Janicic. GCLC--a tool for constructive euclidean geometry and more than that. In Mathematical Software - ICMS 2006, number 4151, pages 58-73, 2006.
[16] K. Kahn, E. Sendova, A.I. Sacristan, and R. Noss. Making infinity concrete by programming never-ending processes. In 7th International Conference on Technology and Mathematics Teaching, Bristol, UK, 2005.
[17] J. Kaput and M. Blanton. Algebraifying the elementary mathematics experience. part I: Transforming task structures. In Chick et al. [6], pages 344-351.
[18] Fou-Lai Lin, Feng-Jui Hsieh, Gila Hanna, and Michael de Villiers, editors. Proceedings of the ICMI Study 19 conference: Proof and Proving in Mathematics Education, volume 1,2, Taipei, Taiwan, 2009. The Department of Mathematics, National Taiwan Normal University.
[19] Filip Maric, Ivan Petrovic, Danijela Petrovic, and Predrag Janicic. Formalization and implementation of algebraic methods in geometry. In THedu'11, volume 79, pages 63-81, Wroclaw, Poland, 2012. Electronic Proceedings in Theoretical Computer Science.
[20] Walther Neuper. Common grounds for modelling mathematics in educational software. Int. Journal for Technology in Mathematics Education, 17(3), 2010.
[21] Walther Neuper. Automated generation of user guidance by combining computation and deduction. In Quaresma and Back [28], pages 82-101.
[22] Tobias Nipkow, Lawrence C. Paulson, and Markus Wenzel. Isabelle/HOL--A Proof Assistant for Higher-Order Logic, volume 2283 of LNCS. Springer, 2002.
[23] R. Noss and C. Hoyles. Windows on Mathematical Meanings: Learning Cutures and Computers. Kluwer, Dortrecht, 1996.
[24] OECD. PISA 2012 mathematics framework. Draft subject to possible revision after the field trial, OECD, 2, rue Andre Pascal 75775, Paris, Cedex 16, France, Nov 2010. www.oecd.org/dataoecd/8/38/46961598.pdf [Retrieved 17.Aug.2012].
[25] John Olive and Katie Makar. Mathematical Knowledge and Practices Resulting from Access to Digital Technologies, chapter 8. Volume 13 of Hoyles and Lagrange [14], 2010.
[26] Seymour Papert. An exploration in the space of mathematics educations. International Journal of Computers for Mathematical Learning, 1(1):95-123, 1996.
[27] Ivan Petrovic, ZoMn Kovacs, Simon Weitzhofer, Markus Hohenwarter, and Predrag Janicic. Extending geogebra with automated theorem proving by using opengeoprover. [1]. to appear.
[28] Pedro Quaresma and Ralph-Johan Back, editors. Proceedings First Workshop on CTP Components for Educational Software (THedu'11), volume 79 of Electronic Proceedings in Theoretical Computer Science. Open Publishing Association, 2012.
[29] Michel Rocard and al. Science education NOW: A renewed pedagogy for the future of europe. Technical report, European Communities, Directorate-General for Research, http://ec.euproa.eu/research/science-society, 2007.
[30] T. Rojano and R. Sutherland. Arithmetic world--algebra world. In Chick et al. [6], pages 515-522.
[31] R.Prank, M.Issakova, D.Lepp, E.T onisson, and V.Vaiksaar. T-algebra - interactive learning environment for expression manipulation. In 7th International Conference on Technology in Mathematics Teaching, volume 1, pages 26-29, Bristol, UK, July 2005.
[32] Bertrand Russell. Introduction to Mathematical Philosophy. George Allen and Unwin, London, 1919. http://philosophyfaculty.ucsd.edu/faculty/ctolley/texts/russell.html.
[33] A. Sacristan, N. Calder, T. Rojano, M. Santos-Trigo, A. Friedlander, and H. Meissner. The Influence and Shaping of Digital Technologies on the Learning - and Learning Trajectories - of Mathematical Concepts, chapter 9, pages 179-226. Volume 13 of Hoyles and Lagrange [14], 2010.
[34] A.I. Sacristan and R. Noss. Computational construction as a means to coordinate representations of infinity. Int.J. of Computers for Mathematical Learning, 13(1):47-70, 2008.
[35] Chris Sangwin, Claire Cazes, Arthur Lee, and Ka Lok Wong. Micro-level Automatic assessment Supported by Digital Technologies, chapter 10, pages 227-250. Volume 13 of Hoyles and Lagrange [14], 2010.
[36] David Tall, editor. Advanced mathematical thinking. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.
[37] H.-G. Weigand. Fiinf Thesen zum Einsatz digitaler Technologien im zukiinftigen Mathematikunterricht. In Mathematikunterricht im Kontext von Realitat, Kultur und Lehrerprofessionalitat, Spektrum, pages 315-324. Springer, Wiesbaden, 2012.
[38] Makarius Wenzel. Isabelle/jEdit a prover ide within the PIDE framework. In J. Jeuring et al., editors, Conference on Intelligent Computer Mathematics (CICM 2012), number 7362 in LNAI. Springer, 2012.
[39] Makarius Wenzel and Burkhart Wolff. Isabelle/pide as platform for educational tools. In Quaresma and Back [28], pages 143-153.
[40] W. Windsteiger. Theorema 2.0: A Graphical User Interface for a Mathematical Assistant System. In Cezary Kaliszyk and Christoph Lueth, editors, Proceedings of UITP 2012 (part of CICM 2012), pages 1-8, 2012.
Walther Neuper
Institute for Software Technology
Graz University of Technology
(1) This paper uses the abbreviation "TP" for two different things without danger of confusion: the respective software products and the respective underlying concepts and technologies.
(2) http://coq.inria.fr/
(3) http://isabelle.in.tum.de/
(4) http://emath.utu.fi/
(5) http://www.risc.jku.at/conferences/cadgme2009/WG/ConvMathAssist.pdf
(6) http://home.pf.jcu.cz/cadgme2010/annotations/ConMAs-10.pdf
(7) http://sites.dmi.rs/events/2012/CADGME2012/mformats.html (eduTPS).
(8)"STEM" is Science, Technology, Engineering and Mathematics.
(9)"CAS" abbreviates Computer Algebra Systems, like Mathematica, Maple, Maxima.
(10)"DGS" abbreviates Dynamic Geometry Systems like GeoGebra, Cabri, Cinderella.
(11) Available at http://poincare.matf.bg.ac.rs/janicic/gclc/
(12) Available at http://www.imped.fi/socos
(13) Available at http://www.ist.tugraz.at/isac/
(14) Exceptions in formula-based mathematics are MathXpert [5], T-Algebra [31] and others.
(15) http://code.google.com/p/flyspeck/
(16) [section]3.2 will come back to "reasoning" with respect to optional, dynamically linked representations.
(17) http://www.geogebra.org/
(18) Building also constructions on Hilbert's or Tarski's axiom systems poses problems not yet solved for automated proofs.
(19) http://isabelle.in.tum.de/dist/library/HOL/
(20) http://isabelle.in.tum.de/dist/library/HOL/Groebne^Basis.html
(21) http://www.ingenious-science.eu/
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In some case, you will like these 2 bedroom loft floor plans. Some times ago, we have collected images to give you imagination, whether these images are inspiring photos. Hopefully useful. Perhaps the following data that we have add as well you need.
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International Data Corporation (IDC) today published an IDC Innovators report profiling four companies offering ground-breaking tools in Talent Intelligence. The four companies are Claro, Eightfold, SeekOut, and Stratigens. IDC defines Talent Intelligence (TI) as the gathering and subsequent analysis of internal and/or external labor data in order to inform talent strategies … [Read more...] about Four Companies Offering Talent Intelligence Platforms Named IDC Innovators
talent management
European HR Solutions Company Alexander Mann Solutions Acquired for £260 million
We have already reported that the HR software market is blazing hot, going through an unprecedented wave of mergers and acquisitions in the last two years. Here is another private equity acquisition - Alexander Mann Solutions, a London-based talent acquisition and management business, has been acquired for £260 million by New Mountain Capital. The HR software market is … [Read more...] about European HR Solutions Company Alexander Mann Solutions Acquired for £260 million


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Commonwealth ICT Ministers Forum will provide an excellent opportunity for participants to learn from global experts as well as share their experiences of successes and challenges in the deployment and utilisation of broadband. This event is a veritable platform to meet with fellow policy makers from across Africa and the wider Commonwealth membership, learn from their experiences and gain insights through interactive sessions.
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Every day, H.J. Baker & Bro., LLC. helps the agriculture industry do more. Our products improve the availability and quality of food. Whether it is feed efficiency for poultry, dairy, swine or aquaculture, the result is the same: more people are fed with fewer resources. The need for food is essential and ever increasing. As an industry and as good neighbors, we will find new ways to feed the world's growing population.
At H.J. Baker, a commitment to excellent customer and supplier relationships is a core part of the company's mission. We focus on growth as a valued producer, marketer and distributor. Our collective skills, industry knowledge and integrity benefit our customers, suppliers, business partners, employees and the community.
Company details
- Address: 2 Corporate Drive, Suite 545, Charleston, West Virginia, 06484
- Country: United States of America
- Tel: +1 203 682 9200
- Fax: +1 203 227 8351
- Email: firstname.lastname@example.com
- Website: https://hjbaker.com/animal-nutrition/
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Startled, the naked girl covered up and tried to send me away but I easily convinced her into having some fun with me by showing her I have a video of her doing some rather naughty stuff.
Even a massive amount of cum is barely enough to cover her chest as she breathes heavily, unable to make a single move.
She continued to stretch her walls and move the toy faster and faster, but her vagina kept gripping just as hard.
Hot as hell.
Description: She lied on her back and spread her legs, letting me eat her out and fuck her pussy with my tongue. Naughty girl next door luna strips off her striped casual dress, revealing her smooth, slender body with delicious perky nipples and beautiful untrimm Met Art.


text:
Friday 27.03.2020
Good morning!
Can you believe we have nearly finished our first week?
You are all making me so proud! I know you are making your adults at home proud too by the incredible messages and photos I'm so lucky to keep being sent, keep it up you superstars! ⭐
It's been really lovely to see photos of some of you enjoying being outside in the beautiful sunshine we have had this week.
So today's challenge is to do something outside, absolutely anything!
(Please make sure you follow the Prime Minister's rules and do this safely).
You could go for a walk, eat your lunch outside, do one of our weekly activities outside, look for some wildlife, absolutely anything you want!
Can't wait to see what you choose to do! ☀
My email address is firstname.lastname@example.com if you haven't used it already.
Enjoy your Friday everyone!
Joke of the day!
Q: How does the ocean say hello?
A: It waves.
Thursday 26.03.2020
Dear children!!!
Well this is a little bit different week! Don`t you think?
I'm going to really miss seeing your faces every day!!
And I would really love to see what you are all up to on your first week at home! Please email me ( firstname.lastname@example.com ) with some pictures that I can add to our class gallery - it is very empty at the moment!
Yesterday I went for a walk in the sunshine around where I live (making sure I listened to the Prime Minister's rules when I was out) and saw lots of beautiful rainbow pictures in people's windows.
They really cheered me up! Have you made one for your window yet?
I would love to see it if you have!
I'm going to make one for my window later.
I'll send you all a photo when it is finished!
Today Emma learnt how to grow sweet peas and daises from seeds !!!
That was a very busy day for her!
Now we are waiting for our tiny seeds to grow!
Today's challenge is a bit of a chilly one...
Can you find a small toy, put it in some water and put it in the freezer?
When the water has frozen and turned to ice, see how you can get the toy back out.
You might want to freeze a few toys separately and use a few ideas to see which way works best to free the toy! Good luck!
Speak to you soon!
Mrs Wagner
Remember now every morning:
9:00 PE with Joe
11:00 Elevenses with David Walliams
From Monday 23rd March and for the next two weeks, SPEEDY SOUNDS films will show at the times below and be available for 24 hours.
Set 1 -9.30 am (GMT)
Set 2-10.00 am
Set 3 -10.30 am
Jokes of the day!
Q: Where do cows go for entertainment?
A: The mooooo-vies!
Wednesday 25.03.2020
Good morning everyone!
What a sunny day we had yesterday! How did you spend your day?
I decided to make something really yummy ...and we baked a bread!
Baking bread is a great project for a six-year-old because YOU can handle the entire process from start to finish with very little help an adult.
(Maybe they want to measure the ingredients, and probably want to handle the oven, but apart from that, there is nothing in bread making that is all that difficult!)
And most important - we all love bread!!!
Another great news is that :
"Every day at 11am, you can listen one of David Walliams ' World's Worst Children stories,
so sit down...take a break...and enjoy 20ish minutes of pure fun!"
Remember now every morning:
9:00 PE with Joe
11:00 Elevenses with David Walliams
and now...
Joke of the day!
Q: Why did the melon jump into the lake?
A: It wanted to be a water-melon.
Tuesday 24.03.2020
Hello Year 1!
It is our second official day of learning from home!
Remember! You can find all the tasks for the week in the "Week 1".
9:00- let`s start with Joe- Tuesday 24th March 2020
9:30 -Have a go at some of the Literacy activities.
10.00 - Stop for a snack, a break and some fresh air!
10.30 - Now have a go at some of the Maths activities.
11.15 - Have a look at some of the online learning links - try out one of the recommended ones before you choose an activity of your choice.
12.00 - Time for a break and some lunch....maybe you could help your mum?
After lunch-Time for phonics- Look for Phonics Folder.
Please try to read daily - 10 minutes!
And remember to practise your weekly spellings everyday.
I have also added some pictures -in Gallery- of our fantastic last day at school!
Hope you have a wonderful sunny day with your families!
Miss you all!
Mrs Wagner
Joke of the day:
What did one plate say to the other plate?
Dinner is on me!
Monday 23.03.2020
Hello my lovely children!
How are you all doing? How was your weekend?
I hope you are all spending time really well together with your families!
Starting at 9.00 am on Monday 23rd March, Joe Wicks is going to be hosting a free daily 30 minute workout aimed at kids LIVE on his YouTube channel, 'The Body Coach TV'.
Let`s burn off some energy every morning!
P.E with Joe Monday 23rd March 2020
Joke of the Day!
Q: Where does the chicken like to eat?
A: At a rooster-ant!
In accordance with the latest government advice, we have had to close some of the school, until further notice. Of course, this is not ideal but with your help, we will endeavour to keep your child's learning moving forward during this unprecedented situation.
On a weekly basis I will post suggestions for activities, projects, websites and videos that we would like your child to access. Frequently, this will involve logging in to the LGFL (London Grid For Learning- https://my.uso.im/ ) suite of websites.
You will have recently received a letter with your child's individual username and password. These login details will remain in place for the duration of your child's time at St Richard's, so please keep them safe and secure.
Go to website: https://my.uso.im/
Look for the red 'USO Login' button (at the top) to enter your username and password.
You may find the following parents' guides to maths and grammar useful:
We understand that each family will be coping with the current situation in different ways, which may affect how much and how often you will be available to support your child with their work, but ideally, we would like each child to attempt at least one activity per day.
To help you to plan for your day, here is a suggested timetable that you could use.
It might help you manage your learning and your family fun.
9:00- let`s start with Joe- daily PE
9:30 -Have a go at some of the Literacy activities.
10.00 - Stop for a snack, a break and some fresh air!
10.30 - Now have a go at some of the Maths activities.
11.15 - Have a look at some of the online learning links - try out one of the recommended ones before you choose an activity of your choice.
12.00 - Time for a break and some lunch....
After lunch-Time for phonics- Look for Phonics Folder and Spellings.
Most importantly, please read with your child on a daily basis.
Read books that you have at home, sharing the reading if your child prefers.
Additionally, Audible are providing audio books for free.
The computer version of 'Teach Your Monster to Read' is free (https://www.teachyourmonstertoread.com/) and helpful with supporting phonics.
A little every day keeps up the momentum.
I will do my best to make the activities fun and to provide some variety, but as this is uncharted territory for all of us, I would welcome any feedback.
You can reach me at: firstname.lastname@example.com with any questions or comments, or to share any work or updates on what your child has been up to.
Please do take care, and I will hopefully see you all sooner rather than later.
-Mrs Wagner


text:
Long bridge
Roman walked on the bridge. When he heard the whistle, he turned and saw running Kamil at the beginning of the bridge. If he went to him, they would meet in the middle of the bridge. Roman, however, rushed and so did not want to waste time returning 150m. He continues walking, and Kamil caught up with him at the end of the bridge. How long was the bridge?
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There are 2743 people working in three workshops. In the second workshop works 140 people more than in the first and in third works 4.2 times more than the second one. How many people work in each workshop?
- Mushrooms
Eva and Jane collected 114 mushrooms together. Eve found twice as much as Jane. How many mushrooms found each of them?


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Crafted from washed twill, the Rylander shoes by Polo Ralph Lauren are an athletic take on the classic boat silhouette. Featuring a branded tongue and a rubber Polo tag, the elegant design has a cork insole for a premium finish and added comfort.
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from Wiktionary, Creative Commons Attribution/Share-Alike License.
- noun music A
dissonantmusic note. Used in Baroquemusic as a decorative embellishment of a melodic note. The grace note is played as quickly as possible and takes no time from the melody note. The symbol for this grace note is a smaller quaver with a line through.
from Wiktionary, Creative Commons Attribution/Share-Alike License
Borrowing from Italian acciacatura.
Sorry, no example sentences found.


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61. Some students are weak in studies. What will be your attitude towards them?
62. If a teacher is not able to answer the question of a pupil he should
63. An individual's ability to learn is
64. How will you demonstrate your impartial behaviour?
65. A teacher who is not able to draw the attention of his students should
Read More Section(Teaching And Research)
Each Section contains maximum 70 questions. To get more questions visit other sections.


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Characteristics of the chiral phase transition in nonlocal quark models
The characteristics of the chiral phase transition are analyzed within the framework of chiral quark models with nonlocal interactions in the mean-field approximation. In the chiral limit, we develop a semianalytic framework that allows us to explicitly determine the phase transition curve, the position of the critical points, some relevant critical exponents, etc. For the case of finite current quark masses, we show the behavior of various thermodynamical and chiral response functions across the phase transition.
- Publication:
Physical Review C
- Pub Date:
- July 2005
- 10.1103/PhysRevC.72.014909
- arXiv:
- arXiv:hep-ph/0410262
- Bibcode:
- 2005PhRvC..72a4909D
- Keywords:
- 12.39.Ki;
- 11.30.Rd;
- 12.38.Mh;
- Relativistic quark model;
- Chiral symmetries;
- Quark-gluon plasma;
- High Energy Physics - Phenomenology
- E-Print:
- 19 pages, 5 figures. Figures 1 and 2 modified, references added, minor changes in the presentation and in the discussion of results. Accepted for publication in Phys. Rev. C


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EP Pro Women's Mock Neck Short Sleeve Golf Polo
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Bring serious style to your next foursome with the EP Pro® Women's Mock Neck Short Sleeve Golf Polo. The unique hexagon print across the front will make this shirt a colorful addition to your wardrobe.
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The undisputed father of the old-school tattoo, Sailor Jerry lived a life worth emulating. He didn't knuckle under to anyone or walk like a lemming into the slow death march of conformity.
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Terrific Tennessee
Visiting Tennessee, a place of both cultural and natural beauty, Barbara Barton Sloane was overwhelmed by affection for a place she had known nothing aboutRead More
Visiting Tennessee, a place of both cultural and natural beauty, Barbara Barton Sloane was overwhelmed by affection for a place she had known nothing aboutRead More
No serious Francophile can afford to miss France's second city, says Jeremy Wayne.Read More
Stockholm offers an unparalleled example of the old and the new.Read More
In her neverending quest to remain a size 4 at 63, Wanderer Debbi K. Kickham revisited Canyon Ranch in the Berkshires, which, she says, takes your health to a whole new plane.Read More
At gilded Blantyre in the Berkshires, you can choose your century, be it 19th or 21st.Read More
"Steel City" has turned the environment into an art form and a competitive sport.Read More
Wanderers Debbi and William Kickham's recent visit to Jamaica including delicious side trips into making jerk chicken and sampling "puddin."Read More
Two new hotels bring new luster to Santa Monica.Read More
Jeremy Wayne visits Sardinia and says he doesn't always like to be beside the seasideRead More
The hills were alive with the sound of … cowbells!Read More
Take advantage of the hot new trend: extended cruising – on Regent Seven Seas ExplorerRead More
It may not be Monaco, Mykonos or Mallorca. But Madeira has quiet charms of its own.Read More
Lisbon had WAG Wanderer Jeremy from "hello" (or was it the natty Borsalino at the hat shop)?Read More
WAG Wanderer Barbara Barton Sloane charts the "road" to happiness on four of the Seven Seas.Read More
WAG Wanderer Jeremy Wayne, an expert traveler himself, offers some unusual tour companies for those who aren't.Read More
Do you know what makes a great trip? When you never want to leave your hotel, because it offers you so much of everything. That is how my sister, my 101-year old mother and I felt about our recent stay in Manhattan at the JW Marriott Essex House New York on Central Park South.Read More
Jeremy Wayne wanders down an English country lane to some of the country's most magnificent homes and gardens.Read More
The Hawaiian words Aloha Aina, Aloha Kai translate to "Love of the Land, Love of the Sea," and nowhere do they sound as true and authentic as on the island of Kauai, the Garden Island.Read More
No need to leave furry or even feathered members of the family home when there are so many pet-friendly hotels.Read More
In the fine city of Cambridge, Massachusetts, there's oh so much to be passionate about.Read More


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The long history of Anvil has been well documented. Formed in Toronto, Ontario, they were on the forefront of a heavy metal explosion in the early eighties but their star never quite reached the heights that so many predicted. Continue reading
Recent Posts
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- How to Survive Your First Heavy Metal Concert?
- The Ghost Inside's Andrew Tkaczyk Recalls Producer Will Putney Pushing Him To Record Drums On Upcoming Album "I'm Glad That's Actually Me Playing"


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Splashes are being heard all over the bay, as water comes crashing down onto Coachman Park! WiLD Splash is here, and now is your chance to grab your tickets to the best Spring Break ever!
Come kick it with your Bay's Party Station at Pinellas Technical College, 901 34th St S. Saint Petersburg, to buy your WiLD Splash tickets without any of the service fees, Wednesday, March 4th from 1pm-2pm.
We will also be giving away a pair of General Admission tickets!
For more information visit the WiLD Splash official page Here


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Lady Agnes Residence
Played 907 times
Lady Agnes Residence game information:
Today Mister Kevin has decided to visit his lady Agnes. Kevin wanted to surprise her in her home but he didn't find her there. Miss Agnes was somewhere out, so Mr Kevin decides to wait for her in the house. He thinks that it would be really nice to take a look of the house in the meantime, seeing the objects there. Let's be a company for Kevin and take a look of the place together.
How to play:
To play this game, you will need only your mouse.
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Helium Gas Tank for Rental and Sale - great if you have your own balloons, or if you need to do your own balloon creations.
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Popularity:#1 of 8 Libraries in Tolland County#8 of 105 Libraries in Connecticut#213 in Libraries
View Address and Phone Number for Tolland Public Library, a Library, at Tolland Green, Tolland CT.
View map of Tolland Public Library, and get driving directions from your location.
Find Genealogy Records related to Tolland Public Library.
Find 6 Libraries within 11.9 miles of Tolland Public Library.
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The Tolland Public Library, located in Tolland, CT, offers books, periodicals, and other information sources to patrons. The library collection also includes electronic databases, films, and other cultural materials. In addition to its collections, the Library also hosts programs and activities in the Tolland County area.You may contact the Library for questions about:


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Wedding Choir needed at Newland, 22/9/18
We are invited to sing at a wedding in Newland Church on Saturday 22nd September. The hymns are Love Divine, Guide me O thou Great Redeemer and One More Step along the World I Go. There is a film song to learn for the signing of the register, 'Turning Page' (from Twighlight IV), easy to find on Youtube, and I have a music edition which is in a key we can sing in more easily than the soundtrack! Please let me know if you're interested (usual contact email@example.com or 01594 530421), but please don't ask for a time just yet - I'll advertise it as soon as I know myself. I'm assuming we could meet early on Tuesdays in Bridges when new term commences in September to practise the song and do any sorting out that may be needed. Meurig
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Looking for an entry level job or internship in biotechnology? This is an overview of how to approach the search and a list of companies that might have relevant entry level and internship opportunities.
The image of a scientist in a white lab coat has some truth to it: lots of opportunities in biotechnology are in research and development, and plenty of them exist for new grads. These positions are available in government agencies, universities, and hospitals. Some positions require advanced degrees or more experience. If you want to pursue a career in research, study biology, biochemistry, or a related field, and pursue an internship or research project before graduating.
Outside of the actual research, you can also find opportunities in the following areas:
|Scientist I, II, or Sr. - Biomedical or Pharma
Experis Vancouver, WA
|Immunology Specialist
Syneos Health Portland, OR
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In 2003, a group of dedicated foster care youth and alumni from across Alaska came together to share their issues and form a Youth Advisory Board. In 2004, the group adopted the name Facing Foster Care in Alaska (FFCA). FFCA is dedicated to sustaining a statewide organization that will continue to work towards supporting foster youth and improving the foster care system.
FFCA has touched the lives of hundreds of youth and alumni across Alaska and the U.S. FFCA members have worked to improve the foster care system on both a state and national level through partnerships with state and private agencies, and youth alumni networks. Recent accomplishments:
More info at: http://www.ffcalaska.org
Facing Foster Care in Alaska
Employer id number (ein)
AddressPO BOX 92644


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Monday 06 April 2020
Elected for London diocese: the Revd Andrew Foreshew-Cain
THE first Church of England incumbent known to have entered into a same-sex marriage has been elected to the General Synod.
He is the Vicar of St Mary with All Souls', Kilburn, and St James's, West Hampstead, the Revd Andrew Foreshew-Cain (News, 27 June 2014). He was among the successful clergy candidates in the diocese of London.
Another priest who has entered into a same-sex marriage, Canon Jeremy Pemberton (News, 17 April 2014), an NHS chaplain, has, however, been unsuccessful in his attempt to represent the clergy of the diocese of Lincoln on the Synod.
The results of the elections began to be announced on Monday. As the Church Times was going to press on Wednesday, 26 of the 42 dioceses had announced all or part of their results.
Fr Foreshew-Cain told The Guardian that he was shocked to have been elected. "I wasn't expecting to get on: I thought the clergy were too conservative to vote for a progressive like me. People would not have voted for me if they didn't want to see the change we represent."
Another openly gay priest from the diocese, the Vicar of All Hallows' by the Tower, in the City of London, the Revd Bertrand Olivier, was also elected, as was the Revd Dr Sean Doherty, a tutor at St Mellitus College, who is married with a wife but also describes himself as same-sex-attracted.
The conservative Evangelical group Christian Concern has condemned Fr Foreshew-Cain's election. In a statement, the group's chief executive, Andrea Minichiello Williams, said: "Andrew Cain's ongoing activism should no longer be tolerated. His actions are designed to undermine the Church and her teaching on marriage. This result should not be recognised, and he should be swiftly removed from church leadership."
The elections co-ordinator for the campaigning group Inclusive Church, the Revd Stephen France, who stood for election but failed to win a seat on the Synod, said that he was optimistic that the group would have increased its numbers once all the results had been announced.
"As of [Tuesday] evening, we have had 46 Inclusive Church candidates elected," he said on Wednesday. "That's about half of the candidates we have put forward. I'm hopeful by the end of the week we will be up to 70."
He dismissed concerns that Fr Foreshew-Cain's prominence could hamper their cause. "It is Christian Concern who are creating that distraction; I don't think Andrew is setting out to do that. He has been duly elected with overwhelming support, and is putting forward his case."
But it was not just supporters of Inclusive Church who have had a good election. The Catholic Group was said on Wednesday to be close to equalling its size during the past quinquennium, even though dozens of dioceses were still to announce their results.
The director of Forward in Faith, Dr Colin Podmore, said: "What makes the achievement even greater is that a lot of Catholic Group members didn't stand again because of age or impending retirement. That meant we had more of a hill to climb, as sitting members have a certain advantage. But those former members are being replaced by younger members of very high quality."
Susie Leafe, who chairs the conservative Evangelical group Reform, has also been returned to the Synod. The group had the support of 31 Synod members during the past quinquennium, but has yet to comment on the results.
Prominent Evangelicals also returned to the Synod include the chairman of the Evangelical Group, the Revd John Dunnett, and Canon David Banting.
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Applications are invited for the position of Bampton Lecturer for 2021.
The Church has a vision of becoming a 'community of communities', with worship at the heart of each of those communities in some form. You will be responsible for developing a coherent strategy and a set of costed proposals which will equip the church to serve communities whose needs are not being met, and who may be marginalised. This will involve scoping the most urgent community needs; helping respond to, and shape, the vision of the congregation; identifying and developing partnerships with other city-centre churches and third sector providers with whom we might wish to work.
Azalea is Christian charity committed to helping all involved in commercial sexual exploitation to find freedom.
This post offers the opportunity to combine responsibility for helping the diocese in its strategic commitment to growing and developing vocations to lay and ordained ministry with part-time parochial ministry in a lovely village setting.
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This Cricketer was mad in Sonali Bendre Love and wanted to Kidnap Her
The relationship between the Bollywood Actresses and cricketers are not hidden from anyone from the world. There have been several Bollywood actresses who have fallen in the love of cricketers and vice versa. In the past, we have seen Anushka Sharma getting married to Virat Kohli. Similarly, several actresses in past have married the cricketers. […]Continue Reading


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The 8 Best Oral Sex Moves You Have Not Tried
Will you be courageous enough to try quantity six?
Securely banishing vegan meals's reputation to be dry and boring forever, 2019's crop of vegan advent calendars are high in indulgent candies to take care of your self or someone you care about from the run up to Christmas – all without the dairy.
Wonder whether chocolate is vegan? Top-notch chocolate brown should perhaps perhaps not include milk; just cocoa butter, that is vegan as it refers and then unwanted fat through the cocoa bean. More companies are creating vegan milk chocolates too, replacing dairy that is traditional for almond or rice versions. Continue reading


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qLearn Receives IDSEI Award For Contribution To Education
The International Direct Selling Educational Institute (IDSEI) honoured qLearn with an IDSEI Award at the V-Malaysia 2019 stage in September. qLearn's IDSEI award was presented by the president of IDSEI, Graham Moore, and acknowledged QNET's contribution to education and advocacy of professionalism in the direct selling industry.
IDSEI is the most dedicated, and the only licensed, international institute that offers globally recognised trademark certification programmes on direct selling and network marketing. They, like us, provide our direct sellers with the education and empowerment they need to practice the direct selling business professionally while executing each step ethically and with confidence.
During his speech, IDSEI President Graham Moore shared with us the current trends in the education sector and the advantages of life-long learning. He discussed how online courses and e-Learning allow for so many more personalised learning options, and on-the-job flexibility. Given the specificity of the subject matter at hand and focus within each course, obtaining micro-credentials has proven to increase learning effectiveness, making e-Learning the favoured global educational option it has become today.
To cap off the qlearn IDSEI award session, our most recent qLearn Graduates marched on stage with their head held high to collect their certificates of completion by V Partner Donna Marie Imson-Lecaroz, Director of International Operations JR Mayer, IDSEI president Graham Moore and QNET Category Manager for Online Education Bernhard Gaksch.
To take your direct selling to the next level, enrol yourself in the Certified Network Marketer (CNM+) course in the QNET eStore. Curated by IDSEI, the CNM+ course will help give you the knowledge and skills you need to become a successful and ethical network marketer. The CNM+ course is a professional certificate programme, and your completion will mean that you receive an official certificate from IDSEI as well as QNET.
As dedicated learners yourselves, make the most of each course QNET has to offer, as they are designed specifically for you, keeping in mind the intricacies of the direct selling industry. Download qLearn and start a course today!
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