Regina Möller
email: regina.moeller@uni-erfurt.de

Peter Collignon
email: peter.collignon@uni-erfurt.de

On the historical development of algorithms -hidden in technical devices?

Keywords: Algorithms, enlightening, fundamental ideas, mathematics education

The content of Adam Ries' book (1574) with the instructions to written arithmetic fostered an enlightening for citizens at this time since they learned how to calculate themselves making it obsolete to pay a "Rechenmeister" for this task. In the early modern times algorithms entered the industrial world by mechanical automatisms intended to take away the routines from people. Nowadays algorithms hidden in many technical devices have become a part of our daily procedures often staying unaware for people. The paper discusses the historical development of the concept of algorithm and its potential for mathematics and general education today.

A historical view on the mathematical aspects of algorithms

The history of algorithms as part of mathematics started in the very early days since the systematic solving of mathematical problems is part of doing mathematics. For example, during the Old Greek times Euclid (300 v. Chr.) gave a rule to calculate the greatest common divisor of two given natural numbers. Another well-known algorithm is the sieve of Eratosthenes, a step-by-step procedure giving prime numbers up to a certain given natural number. Several hundred years later it was Al-Khwarizmi who presented a number of mathematical applications for traders. These were translated into Latin with the title "Algorithmi" containing the artificial word "arithmos" that is derived from "number" and the name of the mathematician (cf. [START_REF] Chabert | Algorithms for arithmetic operations[END_REF].

Hundreds of years later Adam Ries (1574) wrote a booklet on basic arithmetic operations which had great influence on the German citizenship at that time. In the year 1614 the Scottish John Napier published a book on logarithms which made him look like the inventor of logarithms. Leonardo da Vinci (1452-1519) as another significant personality of the Renaissance era demonstrated with the "Vitruvian Man" his understanding of proportions blending geometric descriptions with the Renaissance art. Another hundred years later Leibniz (1646-1716) used the binary code and algorithms on that basis.

In 1815 Augusta Ada Lovelace was the first woman to program for the (technologically underdeveloped) calculator of Charles Babbage (1791Babbage ( -1871)). 1931 Gödel destroyed the dream of many mathematicians that all mathematical theorems can be proved by algorithms by his Incompleteness Theorems: They imply that there is no algorithm with a finite number of steps that can decide if an arithmetic statement is true or false in all cases imaginable. Furthermore, it is impossible to prove consistency for a formal system whose complexity is the same as that of the natural numbers.

In this context we also must accept an incompleteness of arithmetic:

There is no (finite) algorithm that proves the valid statements of arithmetic on the whole. [START_REF] Ziegenbalg | Algorithmen von Hamurapi bis Gödel[END_REF]; translated by the authors).

Later on during the 20 th century von Neumann and Turing provided the mathematical basis for computing with which they started the new era of computer science (Chabert, 1999, p. 457).

During the last decades the development of "artificial intelligence" progressed so rapidly that the nature of algorithms became in immense multiple ways complex (e.g. voice recognition, decision making systems). This development was possible because of rapidly advanced technical conditions which as a consequence offered an improvement in calculating speed and storage space. As an organizational form cloud computing brought enhancements.

Concluding this very short historical view on the successive appearances of algorithms through the centuries it becomes apparent that in former days they were documented in a written form by books with the aim of becoming public to scientists and citizens. Even in the case of military purposes, namely cryptography, the secrecy of the underlying algorithm was not considered appropriate. Only the "key" that gives details of a generaleven publicencryption process had to be hidden, which is the subject of Kerckhoff's principle (Kerckhoff, 1883, p. 12). Nowadays the algorithms behind apps and programs often stay hidden because they are part of a business secret often with the idea to gain money. The question arises when exactly the balance turned from transparency to its opposite and how we can deal with this new situation. One can expect insight from sociological and historical analyses that are not yet been finally concluded.

Today's mathematical exactification and its impact on education

During centuries an algorithm was understood as a series of instructions. The development algorithm's understanding led to distinguished properties.

Steps towards formalization

For the longest algorithms consist of a finite series of instructions to solve a given problem like it was apparent with the sieve of Eratosthenes. Generally accepted are also the following properties which led to a general view and that can be understood as postulates for algorithms [START_REF] Saake | Algorithmen und Datenstrukturen[END_REF]:

(1) The algorithmic procedure must end up with the same outcome in case of the same preliminaries.

(2) Any next step to take is uniquely determined.

(3) The description of an algorithm implies a finite length of the source text which needs to consist of a limited number of signs.

(4) An algorithm is requested to have a limited storage space at any time during its procedure.

(5) An algorithm stops after a finite number of steps.

For didactical purposes it is therefore clear what solving procedures fall under the concept of an algorithm. Like with the concept of sets there is a naïve version of it. And after being confronted with logical difficulties [START_REF] Russell | The principles of Mathematics[END_REF] the need of exactification became apparent and was solved in form of axiomatization. The naïve perception of algorithms has its limits, exactified by the work of Turing and especially of Gödel's Incompleteness Theorems. Both topics, sets and algorithms, finally lead to questions of cardinality and self-reference.

Didactical perspectives for math classes

The idea of an algorithm appears several times in school in mathematics classes. On the elementary level the four basic written arithmetic calculations have algorithmic nature and are named as such in mathematics literature. In our daily practice we do not observe a very detailed reflective attitude; neither teachers nor pupils seem to reflect on their algorithmic nature. At the beginning of the secondary I level the well-known Euclidean algorithm is subject matter in class. Very often this is the first time that pupils get aware of the concept of an algorithm. It is regarded as a prototype of an algorithm since it is named like it.

Following the spiral principle other examples of algorithms occur on the secondary I level (in the German system meaning classes 5 -10: the methods used to solve linear and quadratic equations as well as systems of linear equations. Typical examples on the secondary II level occur with curve sketching, finding extrema, points of inflection and other properties of functions. At this stage the systems of linear equations are again subject matter with an even more algorithmic procedure (Gauß algorithm).

In school algorithms also occur using a calculator or tablets including software for educational purposes. That is for example the dynamic geometry software (DGS; GeoGebra, Cinderella) and Computer Algebra systems (CAS). Nowadays Witzke (2018) uses 3D-Plotters in math classes on different levels to not only visualize mathematical objects but to materialize graphs and geometric solids. This means that algorithms generate didactical tools not only iconical, but even haptical.

Comparing all the mentioned school examples above very few are labeled explicitly as algorithms through the names as the Euclidean and the Gauß algorithm and are therefore more obvious to the pupils. The algorithmic character of other parts of the curriculum is also not easily recognized by the pupils. Even with algorithms whose names prove to be programmatic the teaching normally emphasizes the arithmetic acting and often fails to reflect upon the concept and the meaning of an algorithm. Furthermore, the use of the recently introduced tools in mathematics classes serve didactically as technical devices to manage some type of mathematical problems. However, several of them create some kind of didactical bubble. Some calculators for the secondary level are meaningful only for math classes.

Algorithms as fundamental ideasalibi or truth?

In 1905Felix Klein (1849-1925) claimed functional thinking as a guiding principle for mathematics education which was implemented in the German curriculum through the last decades. Already in the seventies of the last century the importance of algorithmic thinking was claimed (Engel, 1977, p. 5). This was expressed in the light of the situation 40 years ago. Nowadays, there should be no doubt that the situation became much more prominent. The fact that the relevance of algorithms is still not in the center of educational issues is apparent with the "German Bildungsstandards" (Ständige Konferenz der Kultusminister der Länder in der Bundesrepublik Deutschland 2004, 2012): On the primary level algorithms are not mentioned at all and on the secondary I and II levels some of the well-known procedures are classified as algorithms but mostly because of their repetitive character and not in terms of their entire meaning.

We observe a simplified and shortened understanding of the notion on the part of the pupils and teachers as well, in particular relating to its potential in science, society, economy and culture in general. The latter is only recently a subject of scientific research. In the last decades algorithms and their different mechanical realizations have infiltrated our whole life in various and subtle ways. They are omnipresent in every angle of our daily routines. Social networks (like Facebook) and search engines (like Google or Yahoo) are selecting, ordering and evaluating our data, our inputs and their results. Especially the selecting part based on evaluation is a sensitive issue that the companies and institutions have not made transparent, let alone to make an enlightening out of it.

On the contrary, many companies admit that they hide the conditions of their procedures, and one often gets the impression of a conscious deception of the public. Advertisements are individualized simulating or pretending that some kind of care plays a vital role. However the companies gain money by using or selling our data and neutrality and objectivity became illusions.

The role of algorithms for general (math) education

Since our daily life is influenced by algorithms in the ways described above the question arises whether mathematics education can play an enlightening role. The aim here is not only the listing of typical, historically well-known examples, but the task is to elucidate about the significance of the impact on all facets of society. Heymann (2013) emphasized general education in the light of fundamental mathematical ideas. He aimed at showing that the fundamental ideas indicate the universality of mathematics and their relevance for the "entire culture" (Heymann, 2013, p. 158) and this needs to become apparent for the pupils during their school life.

Here we follow the structure of Heymann's frame of reference. The criteria include their relevance for the mathematical development and show the essential nature of mathematics:

(1) Preparation of life;

(2) cultural coherence;

(3) global orientation;

(4) critical use of reason;

(5) sense of responsibility;

(6) communication and cooperation.

There can be no doubt that algorithms affect our daily life in an increasing manner. Therefore, school and especially math classes need to prepare for this fact. Although it is apparent that different countries answer with different attitudes: Europe, primarily the European Union, looks for governmental regulations while the USA care less about rules. The Far East, that is predominantly China, responds to the situation with even stronger governmental rules. In this light cultural coherence as an aim seems to be out-of-date, and, at the same time, the developed global view is an inevitable consequence of it. Hence the critical use of reason becomes more urgent than ever before.

The sense of responsibility needs to be revised and also discussed and reflected in school. And this even more since the situation became much more complex and it is not easily apparent what is for the benefit of humanity and what will harm. Of course communication and cooperation stay important in this light.

Towards an implementation of algorithms in math classes

Both concepts, the set and the algorithm, are simple enough to be presented on the primary level: a set as a Venn diagram (as understood within naïve set theory) and an algorithm as a series of instructions as in a cooking recipe. Therefore both can be represented on the enactive level [START_REF] Bruner | Entwurf einer Unterrichtstheorie (Toward a theory of instruction[END_REF] and are accessible for the iconic one. Their difference lies in the fact that sets can be regarded foremost as a static phenomenon simultaneously due to time, while temporal processing is in the nature of algorithms. In this sense they are complementary. On top of it the concept of sets plays a vital role in the succession of math classes (spiral principle) as the notion of algorithms could and should also [START_REF] Bruner | The process of education[END_REF].

Closely connected is a theory of conceptual progression of mathematical concepts [START_REF] Vollrath | Begriffsbildung als schöpferisches Tun im Mathematikunterricht[END_REF].

Vollrath presents a didactical theory of learning and teaching mathematical terms. He outlines in general what kind of various steps lead to a certain understanding of mathematical notions:

(1) Intuitive level;

(2) content level;

(3) integrated level;

(4) formal level.

On the intuitive level the pupils have an idea of a series of steps as a first access to algorithms. This may be represented by game rules or cooking recipes. On the second level they explore typical properties of algorithms. Actually on the third level a network should be recognized on the part of the pupils. In the case of an algorithm we observe that the notion itself provides a conceptual network. We distinguish two levels of formulation: Firstly, there is an exact description of certain mathematical issues through algorithms represented by flow charts or simple programming languages. Secondly, the notion of the concept of an algorithm itself is to be formulized.

It is apparent that the exactification makes the idea of an algorithm accessible for pupils already during early schooling. It became apparent that the idea of an algorithm is rich enough to be taught on all levels with a shifting emphasis. Therefore mathematics education is obliged to prove again its self-established claim for general education. This needs to be done on the conditions of the presence and how they have developed.

Analysis on consequences for the teaching and learning of algorithms

In early historical times algorithms started as a scientific insight on a mathematical problem and additionally gave information for a systemized solving procedure. Paul Ernest (2004, p. 5) describes it like this:

An example is provided by algorithms. These denote precisely specified sequences of actions, procedures which are as concrete as the terms they operate on. They establish connections between the objects they operate on, and their products. They are a part of the rich structure that interconnects, and thus helps to implicitly define, the terms, and hence the objects of mathematics. This must be relativized to the solver, for what is routine for one person may require a novel approach from another. It is also relative to a mathematics curriculum, which specifies a set of routines and algorithms.

At the very end of the middle ages when Adam Ries, standing at the brink of the abacus using concrete tokens and the decimal number system using symbols on paper, the arithmetic operations in form of a booklet had an enlightening impact on the citizens; an enlightening event that is surely not often recognized in history [START_REF] Deschauer | Das zweite Rechenbuch von Adam Ries[END_REF]). It is an example of a practice through all these times when it was mostly the aim to inform, to get insight and to support people's solution practices.

In order to simplify the solving practices and to avoid mistakes, Babbage, for instance, designed a mechanical machine (1923). And, as soon as machines came into place, the questions also involved avoiding counting errors. Later on the development continued with electronic devices aiming at more and better efficiency which first was the storages place and later the shortening of time.

We observe that the chances of an enlightening process in today's society got at least diminished as soon as commercial ambitions entered the global scene and the hiding of information was part of the business practices. And now our so called informational society is confronted with many different "black boxes".

Here, mathematics and its didactics have a long enough history to feel responsible to undertake the task to clarify the situations and to offer enlightening insights. A view into history confirms once again that we do not observe totally new phenomena. Babbage dealt with a similar situation during the early 19 th century. In his "Reflections on the Decline of Science in England", first published in 1830 [START_REF] Babbage | Reflections on the Decline of Science in England[END_REF], he complained about scientific mishaps: He observed that scientific results were whitewashed: by trimming (levelling of irregularities), cooking (quoting of results which fit to a theory, omitting results that contradict) and forging (inventing of scientific results which fit a conventional meaning or can be fitting to a desired doctrine).

Concluding our observations on the development of algorithms and our analyses on their position today we notice that they bear only minor enlightening character anymore compared to former times.

In order to answer to this situation there needs to be a focus in math classes on prototypes of today's computers. These elementary simplifications could make apparent that the programmed algorithms are combined with money-making commercial ambitions which have to do with evaluating, selecting and, as a result, manipulating. This goes along with Fredenthal's idea of a guided reinvention within his concept of "mathematics as a human activity" [START_REF] Freudenthal | Didactical Phenomenology of Mathematical Structures[END_REF].

We do not favour teaching algorithms as it developed chronologically. This might be an interesting experiment, but would be naïve to push the analogies between ontogenesis and history of science to the limit. Instead of it we support a refinement of historical awareness and instruct students to pay attention to the conditions that caused today's appearance of algorithmsone of several possible implementations of a few fundamental ideas of mankind. This is what Paul Ernest (2015) calls the "postmodern perspective". According to [START_REF] Lyotard | The postmodern condition: A report on knowledge[END_REF], postmodernism can be narrowed to scepticism towards meta-narratives. Since different fields focus on and deal with unlike sometimes divergent ideas and phenomena, their historically developed metanarratives differ enormously among each other. Therefore analysing (deconstructing) metanarratives means to retrace historical developments and the decisions made on the way. This endeavour reveals choices which were taken along the historical evolution within each field.


Jacob J Lamb
email: jacob.j.lamb@ntnu.no

Olivier Bernard

Shiplu Sarker

Kristian M Lien

Dag Roar Hjelme

Perspectives of surface plasmonic resonance optical fibre sensors for anaerobic digestion

Keywords: Hydrogen, biogas, renewable energy, anaerobic digestion, optical sensors, surface plsmonic resonance

Abstract

Biogas production is becoming significantly viable as an energy source for replacing fossilbased fuels. The further development of the biogas production process could lead to significant improvements in its potential. Tte improvement of anaerobic digestion detection technologies is the cornerstone to reach higher methane productivities and develop fully automatized processes to decrease operational costs. New sensors are requested to automatic obtain a better interpretation of the complex and dynamical internal reactor environment. This will require detailed systematic detection in order to realise a near-optimal production process. There is currently a disparity between the complexity of anaerobic digestion, and online detection. By improving the durability, sensitivity and cost of dissolved H 2 sensor technology, further understanding of the anaerobic digestion process may allow the prevention of process failure. Furthermore, the detection of dissolved H 2 directly in the liquid phase would drastically reduce the time for distinguishing an imbalance in the microbial community. The emergence of surface plasmonic resonance (SPR) sensing with optical fibres coupled with the H 2 -sensitive metal palladium, allows detection of dissolved hydrogen in liquid. By implementing these SPR sensors into anaerobic digestion, improvements to the biogas production process, even at small scales, may be achieved by guided the process in the optimum direction, avoiding the collapse of the biological process. This review intends to assess the feasibility of on-line, cost-effective, rapid, and efficient detection of dissolved H 2 in anaerobic digestion by SPR.

2 Introduction

One route to production of renewable, clean energy is through biogas production, where biogas is derived from organic waste materials. The production of biogas is achieved through the anaerobic digestion (AD) of the organic waste materials by various microorganisms [START_REF] Mital | Biogas systems: policies, progress and prospects[END_REF].

Another route to renewable, clean energy is the so-called Power-to-Gas route, where surplus renewable electrical energy is used to produce H 2 by electrolysis. However, H 2 utilisation is facing unsolved bottlenecks related to H 2 transport and storage. An attractive alternative to H 2 distribution is to convert H 2 to methane in a biological methanation process [START_REF] Luo | Simultaneous hydrogen utilization and in situ biogas upgrading in an anaerobic reactor[END_REF][START_REF] Bensmann | Biological methanation of hydrogen within biogas plants: A model-based feasibility study[END_REF]. Methane has a much higher volumetric energy content than H 2 and thus a lower storage transport cost, and can be distributed in existing natural gas infrastructure [START_REF] Gahleitner | Hydrogen from renewable electricity: An international review of powerto-gas pilot plants for stationary applications[END_REF]. By utilising the methanation process, H 2 can be used to reduce the CO 2 in the biogas produced [START_REF] Luo | Simultaneous hydrogen utilization and in situ biogas upgrading in an anaerobic reactor[END_REF][START_REF] Bensmann | Biological methanation of hydrogen within biogas plants: A model-based feasibility study[END_REF]. A promising approach is to input H 2 to the biogas reactor such that the methanation takes place within the reactor (2). However, increasing the concentration of dissolved H 2 in the reactor can inhibit the production of biogas and accumulate VFAs, and in the worst case cause the whole biological process to collapse [START_REF] Spanjers | Instrumentation in anaerobic treatment-research and practice[END_REF].

Due to the complexity of the biogas production process, it is not always straight-forward to determine the state of the AD process from measuring the chemical variables within the digester [START_REF] Mital | Biogas systems: policies, progress and prospects[END_REF]. Despite this, the hydrogen content of produced biogas is a very sensitive indication of imbalance between microbial groups within the digester [START_REF] Steyer | Evaluation of a four year experience with a fully instrumented anaerobic digestion process[END_REF][START_REF] Molina | Selection of variables for on-line monitoring, diagnosis, and control of anaerobic digestion processes[END_REF][START_REF] Boe | State indicators for monitoring the anaerobic digestion process[END_REF]. Although hydrogen is straightforward to measure in the gas-phase, dissolved hydrogen in the liquid phase is considered more appropriate as the liquid to gas mass transfer of hydrogen introduces a significant delay in detection [START_REF] Pauss | Hydrogen monitoring in anaerobic sludge bed reactors at various hydraulic regimes and loading rates[END_REF]. Therefore, dissolved H 2 is recognised as a better approach to an early warning indicator and an excellent complement to VFA monitoring [START_REF] Spanjers | Instrumentation in anaerobic treatment-research and practice[END_REF]. Dissolved H 2 concentrations are, therefore, required control parameters for biogas production monitoring and utilising H 2 for further methanation of CO 2 [START_REF] Luo | Simultaneous hydrogen utilization and in situ biogas upgrading in an anaerobic reactor[END_REF][START_REF] Bensmann | Biological methanation of hydrogen within biogas plants: A model-based feasibility study[END_REF]. It is essential that dissolved H 2 be monitored closely to gain an understanding of the current biological process within the digester.

Cost-effective sensor technologies for monitoring and analysis of H 2 within the reactor are crucial for improving efficiency, productivity and cost of biogas production in many plants.

There are many methods for achieving monitoring of a variety of parameters within bioreactors. These may rely on sample extraction, but in some cases, the sensor can be 5 interfaced directly with the internal environment of the bioreactor. Despite this, there are still improvements required to make ideal sensors for the measurement of dissolved H 2 within the biogas production process. Ideally, a monitoring system must have cost-effective, accurate, specific, stable and sensitive sensors, which can send data to software programs for analysis.

This data can then be computationally analysed, allowing the correlation of the sensor data to a specific parameter or process model. Furthermore, with the increase in small-scale plants, there is a significant demand for sensor technologies that do not have an associated high cost.

In this review, contemporary sensor technologies that can combine AD with on-line sensor technologies focusing on dissolved H 2 are discussed and compared to current industry methods. The contemporary technologies presented have the potential for improving the efficiency, economic costs and productivity of biogas plants, as well as allowing on-line dissolved H 2 monitoring to facilitate H 2 injection for methane upgradation.

Sensors Requirements for State of an Anaerobic Digester Determination

There are distinctive ways that sensors can be fused into the AD plant to satisfy the on-line observing prerequisite. The sensors utilized for estimations of factors inside an AD plant can require the extraction of digestate from the bioreactor, filtration and external sampling (at-line sensor), the sensor to be located inside the digester (in-line sensor), or to be removed from the digester and sampled in a lab (disconnected sensor) [START_REF] Biechele | Sensor systems for bioprocess monitoring[END_REF].

The in-line sensors must be impervious to pressure (somewhat above atmospheric pressure) and temperature (somewhere in the range of 35°C and 55°C), as well as having the capacity to be cleaned. Also, they must be significantly impervious to obstruction by fouling. Ideal sensors must provide proper detection quality, having high precision, specificity and sensitivity. The collected information must then automatically be processed and used in the supervision system. The figured estimations should deliver consistency and be steadiness despite changes in medium changes in biological or chemical contents. In practice, the benefits of the sensor in terms of increasing process reliability and efficiency must be compared to its price, including manpower requirement for human intervention (setup, calibration, cleaning).

6

For live, on-line detection of an AD plant to be valuable, the reaction time of the sensors is a critical trademark [START_REF] Biechele | Sensor systems for bioprocess monitoring[END_REF]. Even though the delay of measurements is explicit to every sensor and application, the time required to acquire an outcome must be small (< 10 minutes) concerning the biological dynamics within the digester. All the more explicitly, the time required to acquire a sensor outcome will depend altogether on the retention time of the AD reactor, duration of analysis and the dead volume of the filtration framework utilised. Also, it must be a lot quicker than the time required for the gathering of H 2 to happen to distinguish and avoid a stress event. With this in mind, the sensor framework must be tailored to the procedure with satisfactory sampling locations. Otherwise, the low effectiveness of the process control will occur [START_REF] Pohlscheidt | Bioprocess and fermentation monitoring[END_REF].

The location of the sensor inside the digester is fundamental, as the digester can be homogeneous or inhomogeneous. This depends mainly on the feedstock type, process size and AD technologies. Also, there are AD methods that have intended inhomogeneous chemical gradients inside the reactor (e.g., up-flow anaerobic sludge blanket (UASB) & fluidised bed reactors), and this is exacerbated with the expansion of reactor size. Despite this, if the digestate is homogeneous, the sensor can be set anywhere inside the bioreactor, and the signal from the sensor will represent the real state; however, if it is significantly inhomogeneous, several sensors might be required to decipher the state of the digester completely.

There are three primary phases (solid, liquid and gas), inside an AD plant. These phases have diverse properties, factors and prerequisites when considering in-line sensors. The liquid phase of the anaerobic digester is an intricate blend of different microorganisms, substrates, nutrients, products, metabolites and dissolved gases. Because of the diffusion rate of components from liquid to gas, it is attractive to make estimations specifically from the liquid phase of the digester. This can decrease the lag time between changes in the AD biological dynamics and identification of these changes.

Traditional H 2 Detection

Dissolved H2 detection

7 Dissolved H 2 produced in AD has long been recognised as an early warning indicator and an excellent complement to VFA monitoring [START_REF] Spanjers | Instrumentation in anaerobic treatment-research and practice[END_REF][START_REF] Pauss | Hydrogen monitoring in anaerobic sludge bed reactors at various hydraulic regimes and loading rates[END_REF]. Additionally, the in situ hydrogen injection into the anaerobic digester is a promising process aimed at increasing the biogas methane content, at the expense of the CO 2 content. It has been hypothesised that adding hydrogen directly into the anaerobic digester may change microbial community composition promoting hydrogenotrophic methanogenesis pathways [START_REF] Bassani | Biogas upgrading via hydrogenotrophic methanogenesis in two-stage continuous stirred tank reactors at mesophilic and thermophilic conditions[END_REF]. This enhances the biological production of methane by 20-40% (2,13), potentially achieving a purity of up to 90% [START_REF] Bassani | Biogas upgrading via hydrogenotrophic methanogenesis in two-stage continuous stirred tank reactors at mesophilic and thermophilic conditions[END_REF][START_REF] Luo | Hollow fiber membrane based H 2 diffusion for efficient in situ biogas upgrading in an anaerobic reactor[END_REF][START_REF] Martin | A single-culture bioprocess of Methanothermobacter thermautotrophicus to upgrade digester biogas by CO 2-to-CH 4 conversion with H 2[END_REF][START_REF] Sarker | Overview of recent progress towards in-situ biogas upgradation techniques[END_REF] when combined with ex situ upgrading. This allows existing biogas plants to be utilised for H 2 , eliminating the need for hydrogen storage (which can be of safety concern) [START_REF] Luo | Integrated biogas upgrading and hydrogen utilization in an anaerobic reactor containing enriched hydrogenotrophic methanogenic culture[END_REF]. Despite this, dissolved H 2 can potentially inhibit production of methane, and in the worst case cause the whole biological process to collapse [START_REF] Spanjers | Instrumentation in anaerobic treatment-research and practice[END_REF]. Moreover, injection of H 2 to a level exceeding a 4:1 stoichiometric ratio between H 2 and CO 2 tends to deplete CO 2 resulting in the rise of pH [START_REF] Luo | Co-digestion of manure and whey for in situ biogas upgrading by the addition of H 2: process performance and microbial insights[END_REF], and inhibiting the autotrophic hydrogenotrophic methanogenesis process [START_REF] Agneessens | In-situ biogas upgrading with pulse H2 additions: the relevance of methanogen adaption and inorganic carbon level[END_REF]. Furthermore, the oxidation of acetate towards methane production is only thermodynamically favourable when the partial pressure of H 2 is below 74 Pa [START_REF] Rachbauer | Biological biogas upgrading capacity of a hydrogenotrophic community in a trickle-bed reactor[END_REF][START_REF] Hattori | Syntrophic acetate-oxidizing microbes in methanogenic environments[END_REF]. Therefore, dissolved H 2 must be monitored continually, especially if H 2 is directly injected into the anaerobic digester for methane upgradation.

Since the critical concentration of dissolved H 2 is only 74 Pa [START_REF] Rachbauer | Biological biogas upgrading capacity of a hydrogenotrophic community in a trickle-bed reactor[END_REF][START_REF] Hattori | Syntrophic acetate-oxidizing microbes in methanogenic environments[END_REF] and difficult to measure, the concentration of dissolved H 2 in anaerobic biogas production is currently calculated from a gas fraction of H 2 detected in the gas phase of the digester. However, due to the limited H 2 mass-transfer coefficient, there is a significant time delay between an increase in H 2 concentration in the liquid and an increase in the gas phase of the digester. Such time delays represent a severe limitation for optimal process control [START_REF] Spanjers | Instrumentation in anaerobic treatment-research and practice[END_REF][START_REF] Pauss | Hydrogen monitoring in anaerobic sludge bed reactors at various hydraulic regimes and loading rates[END_REF]. Extraction techniques designed for off-line dissolved H 2 monitoring are time-consuming and not an option for the rapid process control desired. Other techniques based on the transfer of H 2 from the liquid through a gas-permeable membrane followed by gas phase quantification (gas chromatography or mass spectrometry), are considered too sophisticated and expensive for on-line monitoring.

Therefore, there is a definite need for an effective, reliable, and cost-effective sensor for continuous on-line monitoring of dissolved H 2 in anaerobic bioprocesses.

Gas-Phase Detection

Detection of H 2 gas in the headspace of the anaerobic digester is straight-forward to measure but is delayed significantly due to the mass-transfer coefficient of H 2 . Therefore, the main 8 drawback of the detection in the headspace is that sudden changes in dissolved H 2 levels will not be detected for some time, and, therefore, process failure could occur before any signals of an H 2 increase are detected. Despite this, membrane-covered Clark-type electrodes can be adapted to measure H 2 in the headspace of the reactor, with negligible biofouling of the membrane occurring when in gas-phase. Thermal conductivity sensors like those provided by BlueSens (BlueSens, Germany) can also gauge the concentration of H 2 gas in the headspace.

These sensors use a small heated filament to detect thermal conductivity of the gases in simple gas mixtures, allowing the determination of the percentage of specific gases of interest within a mixture.

Palladium metal oxide semiconductor (Pd-MOS) sensors have also been studied for their suitability for H 2 detection in the gas phase of AD [START_REF] Hörnsten | The use of palladium metal oxide 20 semiconductor structures in quantitative studies of H2 and H2S in processes related to biogas production[END_REF]. The palladium can become palladium hydride in the presence of H 2 , a rapid and reversible reaction that can alter the conductivity of the metal and used to determine the ppm of H 2 . Alternatively, gas chromatography (GC) instruments equipped with thermal conductivity detectors can be used to determine H 2 [START_REF] Boe | State indicators for monitoring the anaerobic digestion process[END_REF].

GC is a very accurate method of H 2 detection in the gas phase and is regularly used as a reference measurement technique for all other H 2 detection methods.

Liquid-Phase Detection

H 2 can be detected using a membrane-covered Clark-type electrode [START_REF] Mišlov | Measurement of dissolved hydrogen concentration with clark electrode[END_REF] in the liquid phase as well. There are extraction and H 2 permeable membrane methods designed for dissolved H 2 monitoring within the digester, but these can be expensive and time-consuming processes.

The membrane requiring Clark-type electrodes can be used for on-line monitoring of dissolved H 2 in the liquid-phase, which relies on the oxidation of H 2 at a Pt electrode and dissociation of molecular hydrogen at a Pd electrode [START_REF] Björnsson | Utilization of a palladium-metal oxide semiconductor (Pd-MOS) sensor for on-line monitoring of dissolved hydrogen in anaerobic digestion[END_REF][START_REF] Huck | Combined amperometric/field-effect sensor for the detection of dissolved hydrogen[END_REF]. Unfortunately, these are prone to biofouling on the membrane surface. Once biofouling occurs, the measurements from this type of electrode are mostly unreliable, and, therefore, the membrane must regularly be replaced to avoid this. So far all reported electrochemical sensors for H 2 detection suffer from membrane fouling resulting in a short lifetime, low selectivity and low signal-to-noise ratio [START_REF] Huck | Combined amperometric/field-effect sensor for the detection of dissolved hydrogen[END_REF]. Although these Clark-type sensors can operate at room temperature, temperature fluctuations must also be taken into account to maintain correct O 2 and H 2 calculations [START_REF] Clark | Continuous recording of blood oxygen tensions by polarography[END_REF].

Measurements of dissolved hydrogen can also be performed using hydrogen microsensors like those provided by Unisense (Unisense A/S, Denmark). Microsensors allow H 2 to diffuse from the internal liquid phase of the anaerobic digestor through a silicone membrane to a 9 platinum anode that is polarized against an internal reference. The current between the palladium electrode and the reference is linearly indicative of the dissolved hydrogen at the tip of the sensor. The current change is in the picoamp range and is measured using a picoammeter. An alternative method for dissolved H 2 detection was described by [START_REF] Björnsson | Utilization of a palladium-metal oxide semiconductor (Pd-MOS) sensor for on-line monitoring of dissolved hydrogen in anaerobic digestion[END_REF]. Here, a liquid-to-gas membrane was utilised to extract dissolved H 2 from the liquid phase, through a Teflon membrane placed in the internal environment of the anaerobic digester. The H 2 was then detected using a Pd-MOS sensor [START_REF] Bjornsson | Utilization of a Pd-MOS sensor for on-line monitoring of dissolved hydrogen in anaerobic digestion[END_REF].

Remarks

The methods outlined here suffer from disadvantages including sample preparation, contamination and biofouling. GC is a real proven alternative at very high accuracy, but requires advanced systems for auto-sampling, transfer of the sample and filtering, and requires a significant amount of time to obtain a measurement. Forms of ultrafiltration are required for liquid phase detection to produce a clear particle-free sample for GC and electrochemical detection methods. This filtration will result in biofouling in the long term, a property that must be considered when using such methods for detection in the liquid phase.

Measurements undertaken in the gas phase of the reactor have limited issues with biofouling of filtration systems; however, the presence of H 2 S in the biogas massively reduces the sensitivity of the sensor systems [START_REF] Hörnsten | The use of palladium metal oxide 20 semiconductor structures in quantitative studies of H2 and H2S in processes related to biogas production[END_REF]. Additionally, the mass-transfer rate of H 2 from the liquid phase to the gas phase is prolonged, resulting in significant delays when detecting H 2 spikes in the AD process [START_REF] Pauss | Hydrogen monitoring in anaerobic sludge bed reactors at various hydraulic regimes and loading rates[END_REF].

Software Sensors

Using mathematical approaches, information provided by on-line sensor systems can be extended. The two strategies for achieving this are based on methods of data analysis and the representation of the dynamics of the plant by mathematical models.

Observers

The incorporation of software sensors with on-line sensor systems allows the prediction of the biological process state by estimating non-measured variables. In the case of dissolved H 2 measurements, estimations of the dominating microbial processes within the reactor can be made. Additionally, software sensor systems are a vital supporting component of closed-loop central strategies. These approaches are also known as state observes or state estimators, and 10 have a large amount of theoretical background. Assimilation of the real-time on-line measured variable(s) can be combined with the theoretical knowledge base of the AD process through a mathematical model allowing the prediction of variables that have a low frequency or no sampling. Depending on the desired accuracy, the frequency of sampling, sensor information quality and frequency of sampling, different strategies for the design of an appropriate software sensor can be applied.

Linear frameworks, or extensions thereof, are popular approaches to software sensing. The most known algorithms for achieving software sensors are based on extended Kalman filters (EKF) and extended Luenberger observers (ELO) (27). Despite their popularity, the required local linear approximation of the process model reduces the certainties of the stability and convergence properties for wide operating ranges. Furthermore, perfect knowledge of the model and its parameters are assumed by these algorithms, allowing the approach to be sensitive to any inaccuracy within the model parameters.

To reduce the dependency of the algorithms on parameter uncertainty, different strategies have been developed by using the mass balance information from the reactor as the base of the estimation scheme. Although mass balance only represents the process partially, it can robustly deal with any missing information. Moreover, estimates of kinetics for some processes in the model can be achieved with the support of gaseous flow rates using asymptotic observers [START_REF] Bastin | Adaptive control of bioreactors[END_REF]. These methods can achieve a linear observer by relying on changes of variables cancelling nonlinear terms within the process [START_REF] Bernard | Software sensors to monitor the dynamics of microbial communities: application to anaerobic digestion[END_REF]. However, the dilution rate controls the convergence rate when utilising asymptotic observes. Simultaneous estimating of model parameters and the state of the process can be performed with adaptive observers [START_REF] Rodríguez | An adaptive observer for operation monitoring of anaerobic digestion wastewater treatment[END_REF]. Alternatively, using the process model and its known parameters allows the tailoring of nonlinear observers with the nonlinearity of the process accounted for by the algorithm [START_REF] Flores-Estrella | H∞ control of anaerobic digester for winery industry wastewater treatment[END_REF]. A significant limitation of the implementation and calibration of such observer algorithms is their inherent complexity.

Internal observers offer a flexible method by providing guaranteed estimate intervals where the state of the variable lies, avoiding reconstruction of precise numerical values [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF]33). This has been observed successfully for use in AD [START_REF] Bernard | Closed loop observers bundle for uncertain biotechnological models[END_REF][START_REF] Alcaraz-González | Application of a robust interval observer to an anaerobic digestion process[END_REF]. Due to the uncertainty of concentrations in the influent, this parameter is inherently difficult to use to estimate the process accurately. To overcome this, an algorithm that simultaneously estimates the influent concentrations using both input and state observers has been shown to alleviate this drawback 11 [START_REF] Jáuregui-Medina | Observer-based input estimation in continuous anaerobic wastewater treatment processes[END_REF]. Further development of these observers has resulted in their extension to handle spatial distributions within the anaerobic digester [START_REF] Delattre | Observability analysis of nonlinear tubular (bio) reactor models: a case study[END_REF][START_REF] Aguilar-Garnica | A multivariable control scheme in a two-stage anaerobic digestion system described by partial differential equations[END_REF].

Emerging SPR Sensor Technologies

More recently, various optical sensor principles have been proposed for H 2 monitoring [START_REF] Jiang | Highly sensitive dissolved hydrogen sensor based on side-polished fiber Bragg grating[END_REF][START_REF] Mak | Optical fiber sensor for the continuous monitoring of hydrogen in oil[END_REF][START_REF] Westerwaal | Fiber optic hydrogen sensor for a continuously monitoring of the partial hydrogen pressure in the natural gas grid[END_REF][START_REF] Yan | A fast response hydrogen sensor with Pd metallic grating onto a fiber's end-face[END_REF][START_REF] Zhang | Recent advancements in optical fiber hydrogen sensors[END_REF][START_REF] Ohodnicki | Pd/SiO2 and AuPd/SiO2 nanocomposite-based optical fiber sensors for H2 sensing applications[END_REF][START_REF] Perrotton | A reliable, sensitive and fast optical fiber hydrogen sensor based on surface plasmon resonance[END_REF][START_REF] Wadell | Drift-corrected nanoplasmonic hydrogen sensing by polarization[END_REF][START_REF] Coelho | Fiber optic hydrogen sensor based on an etched Bragg grating coated with palladium[END_REF][START_REF] Downes | Optical Fibre Surface Plasmon Resonance Sensor Based on a Palladium-Yttrium Alloy[END_REF][START_REF] Luo | Research on High Sensitive D-Shaped FBG Hydrogen Sensors in Power Transformer Oil[END_REF][START_REF] Ma | Pd/Ag coated fiber Bragg grating sensor for hydrogen monitoring in power transformers[END_REF][START_REF] Uddin | Dissolved hydrogen gas analysis in transformer oil using Pd catalyst decorated on ZnO nanorod array[END_REF]. Sensors based on the H 2 uptake in palladium appear particularly promising. The volume expansion of palladium films or nanoparticles caused by metal-hydride formation can be measured very accurately using optical techniques [START_REF] Jiang | Highly sensitive dissolved hydrogen sensor based on side-polished fiber Bragg grating[END_REF][START_REF] Mak | Optical fiber sensor for the continuous monitoring of hydrogen in oil[END_REF][START_REF] Ohodnicki | Pd/SiO2 and AuPd/SiO2 nanocomposite-based optical fiber sensors for H2 sensing applications[END_REF][START_REF] Coelho | Fiber optic hydrogen sensor based on an etched Bragg grating coated with palladium[END_REF][START_REF] Luo | Research on High Sensitive D-Shaped FBG Hydrogen Sensors in Power Transformer Oil[END_REF][START_REF] Ma | Pd/Ag coated fiber Bragg grating sensor for hydrogen monitoring in power transformers[END_REF]. Additionally, the surface plasmon resonance (SPR) of gold (when coupled to palladium in a layer stack), can be measured accurately, with the plasmon resonance frequency being modulated by the hydrogenation of the palladium layer [START_REF] Zhang | Recent advancements in optical fiber hydrogen sensors[END_REF][START_REF] Perrotton | A reliable, sensitive and fast optical fiber hydrogen sensor based on surface plasmon resonance[END_REF][START_REF] Downes | Optical Fibre Surface Plasmon Resonance Sensor Based on a Palladium-Yttrium Alloy[END_REF]. SPR sensors exploit the interaction of incident electromagnetic waves with the oscillation of the plasmon wave of a material (e.g., gold). When the momentum of incident electromagnetic waves matches the momentum of the surface plasmon waves in the gold, the electromagnetic waves are coupled to the plasmon wave and are diminished (FIG? ??). The frequency of the plasmon wave can be altered by changes in the materials used, therefore changing the wavelength of incident electromagnetic waves absorbed. An example of this is the use of a bare, cladding-less optical fibre with a gold and palladium layer-stack (Fig) . The incident electromagnetic wave interacts with the plasmon wave on the surface of the gold layer.

Furthermore, this plasmon wave can be modulated by changes in the permittivity of the gold.

By adding an H 2 -sensitive layer of palladium, the formation of palladium hydride effects the electrical field of the gold, changing its permittivity, and changing the momentum of the surface plasmon resonance of the gold. The interaction of the incident electromagnetic waves with the gold will be changed, causing different electromagnetic waves to be diminished.

Using spectroscopic methods, this will culminate in a shift in apparent absorption of electromagnetic waves as a consequence of H 2 presence.

In order to engineer an SPR-based sensor with palladium as the sensing material for use in the liquid phase of an anaerobic digester, the thicknesses of the gold and palladium layers, and the incident angle of the light to the layers must be considered. Using a straight-forward model accounting for the refractive indexes of the silicon optical fibre, gold, palladium and water, a considerable variation in sensor ability is observed as a function of the incident angle 12 (FIG). This is also apparent with varied thicknesses of the two metal layers (FIG). To achieve a dissolved H 2 sensor with high sensitivity and high signal using only gold and palladium metal layers, an incident angle of XX, with metal thicknesses of XX of gold and XX of palladium is required. Although the thickness of the metal is straight-forward to achieve with sputtering deposition techniques, the angle of incidence requires further design considerations.

The metal layers can be deposited on the side surface of the fibre. Increases in the incident light angle can be performed on such designs by placing the optical fibre in a d-loop (Fig) , or by using a Bragg grated fibres (Fig). This will allow the incident light angle to be optimised for coupling of the electromagnetic wave to the SPR wave of the gold layer. Alternatively, the metal layer stack can be deposited on the end-tip of a fibre, acting as a mirror (Fig) . With the use of a multimode fibre, this will allow the electromagnetic wave to achieve coupling with the SPR wave of the gold layer. Furthermore, the metal layer stack could be deposited onto a transparent patch and installed on the internal space of the reactor behind a glass window.

Probing of the sensor patch would then be achieved using an external optical fibre, reducing complexity and cost.

Calibration of SPR Sensors

With using palladium as the sensing material, the selectivity for hydrogen is very high.

Therefore, calibration can be achieved straightforwardly by use of GC measurements of hydrogen concentration in the liquid phase. Before a palladium-based SPR sensor can be utilised, the consistency and accuracy of the sensor must be confirmed. Different SPR sensor datasets must be comparable to establish successful data analysis. A variety of pre-processing methods can be applied to the data sets to achieve this. Some examples include deviation [START_REF] Sivakesava | Simultaneous determination of multiple components in lactic acid fermentation using FT-MIR, NIR, and FT-Raman spectroscopic techniques[END_REF], filtering (53) normalisation, standardisation, centring, weighing and scaling [START_REF] Rinnan | Review of the most common pre-processing techniques for near-infrared spectra[END_REF] of the data sets. Pre-processing is a powerful and sensitive component in data analysis [START_REF] Lourenço | Bioreactor monitoring with spectroscopy and chemometrics: a review[END_REF], where consideration of the chosen method is required depending on the type of data used. The exclusion of irrelevant data is also a pre-processing method that is essential. This can be achieved by normalisation or baseline subtraction/correction. Matching of the data set to known dissolved H 2 concentrations determined by GC can then be performed.

Promising Perspectives of SPR Sensors in AD

SPR sensors are advantageous as they can also be applied to other variables revealing the internal state of the complex anaerobic process. Acetic acid is a VFA that can be produced during certain stages within the AD of biomass. At high concentrations, it will inhibit the activity of many microorganisms within the digester (REF NEEDED). Using SPR technologies, measurements can be made of acetic acid with the receptor dye aluminium phthalocyanine chloride deposited on a layer of gold [START_REF] Evyapan | Study of the sensor response of spun metal phthalocyanine films to volatile organic vapors using surface plasmon resonance[END_REF]. This design has been shown to sense acetic acid in a liquid environment [START_REF] Evyapan | Study of the sensor response of spun metal phthalocyanine films to volatile organic vapors using surface plasmon resonance[END_REF], suggesting its adaptation into AD may provide an accurate only measurement of acetic acid.

By utilising colloidal copper nanoparticles, detection of ammonia concentrations have been observed utilising SPR [START_REF] Prabhash | Localized surface plasmon resonance based highly sensitive room temperature pH sensor for detection and quantification of ammonia[END_REF]. By suspending colloidal copper nanoparticles in a solgel on a transparent disk behind an observation window within the digestor, direct measurements of ammonia levels would be possible by external probing of the sensor disk with an optical fibre.

Methane concentration is usually measured in the gas-phase of the digester; however, it is also possible to measure this variable within the liquid-phase of the digester. By depositing a layer of modified polydimethylsiloxane (PDMS) polymer that contains cryptophane-A molecules as the receptor, on top of a layer of gold, methane may be detected using SPR [START_REF] Boulart | A novel, low-cost, high performance dissolved methane sensor for aqueous environments[END_REF]. This detection is due to the selective, yet reversible, affinity of cryptophane-A to methane.

The H 2 S must be removed from the exhaust gases by bio-filtration methods. It is imperative that the post-filtration gas has no H 2 S component, and therefore, this gas must be monitored continuously. Titanium oxide is known to react with H 2 S, producing titanium sulphide. This reaction has been observed to cause a change in the resonance frequency of the surface electrons of silver [START_REF] Mishra | Surface plasmon resonance based fiber optic hydrogen sulphide gas sensor utilizing nickel oxide doped ITO thin film[END_REF]. The resonance can then interfere with incident light, as a function of the H 2 S concentration in the gas. Although these variables have been detected using developed surface plasmon detection techniques, they are all yet to be adapted for utilisation in an anaerobic digester.

Perspective and Challenges for the Developments of SPR

The main limitations for palladium-based SPR sensors detecting dissolved H 2 in an AD are palladium poisoning by H 2 S, biofouling and the chemical complexity of the digestate.

Moreover, the repeatability of the data from the sensor in a specific digestor may change as a result of influent changes, and the comparability between different digestors with different feedstocks may also be a challenge. This can be due to a variety of reasons were certain unknowns within a complex mixture affect the dissolved hydrogen detection of the palladiumbased SPR sensor. H 2 S is produced in small quantities in AD and has a negative effect on the sensitivity of palladium to H 2 . Interferences of H 2 S to palladium is partly reversible, but to some extent, it is irreversible [START_REF] Hörnsten | The use of palladium metal oxide 20 semiconductor structures in quantitative studies of H2 and H2S in processes related to biogas production[END_REF]. This has to be considered when developing a mathematical model for the sensor, as the level of poisoning will have some dynamic trend due to H 2 S poisoning being partly reversible. Sample preprocessing is also possible by significantly increasing the pH of the sample to remove H 2 S; however, this is not desirable as it adds another step to the detection of H 2 .

As with many sensor systems placed inside the liquid phase of an anaerobic digestor, biofouling is a real limitation. There are plenty of methods for filtration of samples available that significantly reduce biofouling on the sensor at the cost of filter biofouling. An alternative is to utilise thin non-stick films of XXX on top of the palladium layer, to limit fouling of the sensor. Additionally, by allowing the sensor patch to be replaceable, the sensor could easily be replaced routinely without replacing any of the other components within the sensor setup. Despite this, the selectivity of palladium for H 2 in such a complex mixture allows many variables within the digestate to be ignored by this sensor. Therefore, the ability for the sensor to detect a rapid rise in dissolved H 2 (irrespective of providing an entirely accurate concentration measurement), would allow rapid altercations to be made to avoid biological inhibition and potential biological process collapse. This will give the controller of the plant valuable time to rectify the biological process, by eliminating the delay in H 2 detection in the gas phase due to the slow mass transfer of H 2 between liquid and gas phases.

Optically-based SPR sensors integrated directly into the AD would result in rapid on-line dissolved H 2 detection within the liquid phase. Direct integration of these sensors (whether directly placed within the reactor, or behind an observation window), and negligible time 15 required to achieve sensor-H 2 interactions, will allow instantaneous measurements of the dissolved H 2 within the anaerobic digester. The development of an on-line sensor system based on SPR sensors for detection of dissolved H 2 for use in AD, coupled with automated post-treatment of the sensor data sets is a realistic alternative to traditional H 2 detection methods as.

Concluding Remarks

Current AD plants require sensor advances that permit the estimation of dissolved H 2 on-line, not only in large-scale plants but in small scale plants also. These advances will give the operators of AD plants the possibility to acquire not only a higher comprehension of the perplexing dynamics inside the AD plant but additionally foresee the heading of the reactor.

Traditional variable estimations required extensive and sophisticated instrumentation.

Additionally, with difficult extraction and preparation, experts are required leading to increased expenses. These costly, for the most part, off-line procedures are highly accurate, albeit typically do not depict an altogether precise display of the interior AD plant condition due to the potential time delay in obtaining the information. New sensor frameworks for detecting the internal dissolved H 2 condition of an AD plant must most likely show high sensitivity, specificity and precision while keeping up the practicality of an on-line information stream.

New sensor frameworks may expect parts to be disposable (e.g., SPR sensor patches); however, they must be able to achieve comparable detection attributes between disposable parts. The lifetime of such sensor segments like the sensor patches might be short in traditional terms; however, must be low-cost so they can be replaced as required. These sensors must be simple, small and isolated to encourage this. SPR optical-based sensor advances for liquid phase dissolved H 2 detection can fulfil the necessities as future on-line biogas sensors, and will fundamentally avoid the identification delay as seen with traditional gas-phase detection.

  Figures

Acknowledgements ....................................................................................................................... 16 Figures ...

........................................................................................................................................ 17 References ..................................................................................................................................... 18 Acknowledgements Jacob Lamb acknowledges the support from the ENERSENSE research initiative, and his research was supported by funding from the Norwegian University of Science and Technology -NTNU, and the Norwegian Research Council. Olivier Bernard gratefully acknowledges ENERSENSE for supporting his sabbatical stay at NTNU.


Bruna Moustapha-Corrêa

Aline Bernardes
email: aline.bernardes@uniriotec.br

Victor Giraldo
email: victor.giraldo@ufrj.br

Historical tasks to foster problematization

Keywords: Commognitive conflict, history in teacher education, mathtask, Ahmes' Papyrus

 and Kjeldsen & Petersen's approaches (2014). More specifically, we discuss how we used history to design the historical task of the first task round, based on the so-called Aha problems from Ahmes' Papyrus. History of mathematics is used in the problematizing activities to move teachers out of their comfort zone and to legitimate different ways of producing mathematical knowledge. Some episodes are presented only to illustrate our discussion.

Introduction

How do teachers deal with their students' unexpected solutions? We believe that acknowledging and legitimating different kinds of mathematical reasoning in all educational levels is a crucial aspect of teachers' knowledge and professional activity, and, therefore, this discussion cannot be put aside in teacher education programs. With this motivation, we have investigated how teachers deal with students' different solutions for mathematical problems in lower and upper secondary schools.

The experience reported in this paper is part of the doctoral research of the first author, jointly supervised by the other two authors. The aim is to design activities to problematize teachers' understandings of mathematical knowledge production, in particular their own mathematical knowledge production, and how they legitimate (or not) students' mathematical knowledge production. We understand problematizing as considering different alternatives for something that is usually taken for granted, which opens ways to explore questions that are usually not even regarded as relevant. In this paper, we focus on the use of historical problems and contexts to trigger problematization. For this, we consider recent trends of the historiography of mathematics, which builds upon the perspective of social and cultural contexts within which mathematical practices are produced. Our research combines frameworks on the integration between history of mathematics and teaching of mathematics and on teachers' education.

From the historical framework, we draw from Kjeldsen and colleagues' (e.g [START_REF] Kjeldsen | Bridging history of the concept of a function with learning of mathematics: Students' metadiscursive rules, concept formation and historical awareness[END_REF] work, which suggests that the history of mathematics is a source of discourses governed by different metarules. We believe that history can be used to legitimate unusual practices in specific contexts. From the teacher education framework, we consider the importance of a kind of knowledge which is developed through school practice [START_REF] Tardif | Saberes profissionais dos professores e conhecimentos universitários: elementos para uma epistemologia da prática profissional dos professores e suas consequências em relação à formação para o magistério[END_REF]. For this reason, the participants in our study are experienced teachers, who were taking a professional master's program1 . Our field work took place in a module, which was part of their degree. In the sessions, participants were invited to analyse solutions for mathematical problems from Brazilian school syllabuses, including students' strategies and historical approaches. This was done by combining tasks inspired by [START_REF] Biza | Using tasks to explore teacher knowledge in situationspecific contexts[END_REF] and Kjeldsen and Petersen's history approach (2014). The field work's methodology was designed in the light of commognitive framework [START_REF] Sfard | Thinking as Communicating: Human development, the growth of discourses, and mathematizing[END_REF], which will also be used in data analysis. The aim of this paper is to discuss the use of history in the design of these tasks. Episodes from the sessions are reported only to illustrate our task design.

Theoretical Background

According to Sfard, learning is understood as participating in a discourse and mathematics is a kind of discourse. For the author, discourse is a "special type of communication […] distinguishable by their vocabularies, visual mediators, routines, and endorsed narratives" (Sfard, 2008, p. 297). She adds: "In mathematics, endorsed narratives are those that constitute mathematical theories." (idem, p. 298) However, Sfard does not reduce narratives to formal written texts. For her, narratives correspond to a series of spoken or written utterances, whilst routines grasp a set of metarules. These rules are related to the actions of discursants and are historically established, changing through the different discourses. Sfard (2008, p. 260) also points out that commognitive conflict appears when one encounters a discourse incommensurable with one's ownwhen familiar routines are confronted with other people's alternative ways of implementing the same discursive tasks, grounded in different metarules.

Commognitive conflicts are at the core of our research. Thus, to lead participants to discuss and reflect on different ways that mathematics can be produced, we expose them to different discourses in two ways: using historical sources, and through tasks inspired by mathtasks [START_REF] Biza | Using tasks to explore teacher knowledge in situationspecific contexts[END_REF]. [START_REF] Biza | Using tasks to explore teacher knowledge in situationspecific contexts[END_REF] propose mathtask as a situation-specific task designed to engage teachers with classroom situations, to which they are invited to discuss attitudes they would take in the actual classroom. These situations are usually hypothetical, yet data grounded and likely to occur in the actual practice. The connection between history of mathematics and the commognitive framework is inspired by Kjeldsen's work. Since history is a source of discourses governed by different metarules (e.g. [START_REF] Kjeldsen | Bridging history of the concept of a function with learning of mathematics: Students' metadiscursive rules, concept formation and historical awareness[END_REF], it provides different ways of approaching problems and their solutions. Arcavi and Isoda (2007, p. 112) highlight the importance of teachers' listening capabilities, that is, "giving careful attention to hearing what students say (and to see what they do), trying to understand it and its possible sources and entailments". This is close to our goal, since we aim to problematize teachers' criteria to legitimate (or not) students' production of mathematical knowledge. In our work, history is used to reach this problematization by discussing historical solutions to mathematical problems, which may differ from the usual strategies teachers are familiar with. Thus, the problematizing activities we propose are designed to use historical sources that move teachers out of their comfort zones, by revealing consistent practices within specific social contexts.

The Problematizing Activities Design

The problematizing activities we propose are organized in task rounds, each one consisting of three stages: triggering task, historical task, and reflections on teaching. Our field work was conducted in a module of a professional master's program for mathematics teachers. Participants were 12 mathematics experienced teachers (who teach students with ages range from 11 to 18) who were taking this module. The module comprised thirteen 3-hour sessions (organized in three task rounds), including students' assessment. In this paper, we discuss how we used history, in articulation with frameworks on teachers' education, to design the activities, with a special focus on the historical task. We address the case of one of the three task roundsbased on the so-called Aha problems from Ahmes' Papyrus. Some episodes are presented, in order to illustrate our discussion. It is not an aim of this paper to rigorously analyse this empirical data.

The task rounds' structure

Each round starts with a triggering task, in which participants are given hypothetical student' solutions to a mathematical problem. The aim is to bring about the ways they deal with students' responses, and to which extent they are open to unusual solutions. This is followed by a historical task, in which participants are presented to excerpts from original sources. The aim is to shake participants' comfort zones formed by the kind of solutions they are familiar with, since the historical discourses are usually different from the ones used in the classroom. In the last stage of each round, reflections on teaching, participants are again given hypothetical student' solutions, which now also draw upon historical solutions. The structure of the triggering task and the reflections on teaching stages are inspired by [START_REF] Biza | Using tasks to explore teacher knowledge in situationspecific contexts[END_REF] mathtask model, namely: firstly, participants are asked to solve a mathematical problem; then they are asked to analyse student solutions to this problem, and to describe the feedback they would give to students in an actual classroom situation. In each of the three stages, participants are first asked to write down individual responses, and then invited to collective discussion. Either by bringing historical excerpts, or by using mathtasks, we try to promote a commognitive conflict [START_REF] Sfard | Thinking as Communicating: Human development, the growth of discourses, and mathematizing[END_REF]. Our goal is to encourage the participants to reflect and discuss on: i) their own routines regarding their classroom practices, ii) how they understand mathematics, and also iii) the criteria they apply when judging their students' mathematical solutions and reasonings. For the choice of each task round's topic we take into account: potential to allow different solutions; relevance to the participants' practice; and availability of primary and secondary historical sources.

The first task round: context and structure

Our first task round is built around the simple false position rule, according to which the solution is reached by using an "experimental number" that is adjusted by a factor. Thus, the process of finding the solution is not based on a representation for the solution, as in the case in the contemporary algebraic methods. We choose the context of the ancient Egyptian Aha problems (e.g. [START_REF] Roque | História da Matemática: Uma visão crítica, desfazendo mitos e lendas[END_REF][START_REF] Chace | The Rhind Mathematical Papyrus: British Museum 10057 and 10058[END_REF] from Ahmes Papyrus to explore different solutions to problems modelled by linear equations nowadays, as a potential trigger of commognitive conflicts. We could consider many metarules regarding the discourse from Aha problems (e.g. the ones related to the Egyptian symbols, the ones related to the structure of the Aha problems, the ones related to the Egyptian mathematical practices), but we focus on the metarule related to the assumption of an "experimental number" to start the Aha problemwhich considerably differs from representing the unknown quantity by a letter and operating with it. Thus, we use the simple false position rule as a means of problematizing algebraic solutions commonly used at school. The choice of this activity is also justified by the fact that the false position rule is grounded in the notion of proportionality. Despite this is a key concept in school mathematics, it is sometimes underestimated in the teacher education programs (at least in the Brazilian context). The triggering task consists of a mathtask with two hypothetical solutions to the following problem:

Marina likes to make problems for her father inspired from what she learns at school. One night she said: "Dad, find out how many Reais2 I have! The tip is: if I add a quarter of what I have to what I have, I'll get R$15,00."

Ten participants attended this session, all of which presented an algebraic solution, as expected. A mathtask with two students' solutions to the problem was then discussed. The mathtask's context was an introductory lecture on algebra at elementary school for 12/13 years old students. The first solution uses a diagram to split the whole in 4 parts, and then add a fourth one. So, 15 is divided by 5, and the result is multiplied by 4. The second solution is a "trial and error" strategy. Aiming to set up commognitive conflicts, both student solutions were designed considering routines different from the algebraic solutions. The metarules behind the first solution's routine are related to the use of fraction bars, whilst the ones in the second solution routine are related to the use of a trial and error strategy.

Designing the historical task: our methodological option

The historical task concerns two well defined moments: an immersion in original sources and a historical overlook. These approach were inspired by [START_REF] Arcavi | Learning to listen: From historical sources to classroom practice [Special Issue[END_REF]'s hermeneutic method. The authors propose a sequence of tasks aiming to parsing the source, posing questions to oneself (or to a peer) around it, paraphrasing parts of the text in our words and notations, summarizing partial understandings, locating and verbalizing what is still to be clarified, and contrasting different pieces for coherence. (Arcavi & Isoda, 2007, p.116) Thus, immerging in a source is the opportunity to analyse and understand different mathematical practices, contrasting them with the participants' own ones. In the historical overlook, we discuss (mathematical and cultural) contexts of the source and some aspects of the Egyptian mathematics. We adopt this approach as a means to lead participants to experiencing "other perspectives", as [START_REF] Arcavi | Learning to listen: From historical sources to classroom practice [Special Issue[END_REF] posit, which, in the context of the history of mathematics, may be the ones from the primary sources, as we used in this research. As these authors suggest, in the context of the classroom, the "other perspective" is the students' ones. So, they argue that trying to understand historical sources may lead teacher to be more receptive to students' productions, specially the unexpected ones.

The aim of the first round was to explore Problem 25 of the Ahmes' Papyrus, which states: "A quantity whose half is added to it becomes 16." Firstly, we invited participants to analyse Problem 25 in Hieratic and Hieroglyphic (Figure 2), as it appears in the Ahmes' Papyrus [START_REF] Chace | The Rhind Mathematical Papyrus: British Museum 10057 and 10058[END_REF]. Next, they were presented to Problem 25, as it appears in Figure 3, with translation to English and to (our) Portuguese. After presenting a brief contextualization on Ancient Egypt, the role of the scribes in the society organization, and what Ahmes' Papyrus and, particularly, the Aha problems are about, we presented Problem 25 again as it appears in Figure 3, and invited participants to explore it. We then proposed tasks from Arcavi and Isoda's work, including: the elaboration of a dictionary of Egyptian symbols, especially numbers; a problem with Egyptian multiplication; and a set of tasks exploring Problem 243 [START_REF] Arcavi | Learning to listen: From historical sources to classroom practice [Special Issue[END_REF] in stages. Finally, we discussed Ancient Egyptian mathematics, especially number system, fractions and operations, using [START_REF] Roque | História da Matemática: Uma visão crítica, desfazendo mitos e lendas[END_REF] as the main reference. We then resumed Problem 24's solution and encouraged the participants to understand the solution as a whole. As Arcavi and Isoda suggest, we asked them to try to present a solution using a different experimental number. Moreover, we invited them to try to compare the solutions for Problems 24 and 25. The solution was written in modern terms and the expression "false position rule" was presented. Finally, the questions (inspired by [START_REF] Winicki | The Abalysis of Regula Falsi as an Instance for Professional Development of Elementary School Teachers[END_REF] were posed to guide the discussion: 1) In modern terms, to which kind of equations do the Aha problems correspond? 2) Present a justification for the false position rule. 3) In the false position rule, does the result depend on the experimental number? 4) Does the false position rule work for all kinds of Aha problems? Why? 5)Why do you think such a rule was invented? 6) Why is it anachronic to claim that the Egyptian "solved equations"?

To conclude the historical part, we asked the teachers about simple false-position-rule rationale, that is, to explain why the method is valid. After a brief discussion, we presented a modern justification, using a linear function's graphic in order to show the proportionality between the experimental number (false position) and the solution. The reflections on teaching stage was guided by a mathtask presenting a situation in which a 12-year-old boy that doesn't know algebra yet solves Problem 25 by trial. His father wants to convince him that it is easier to solve the problem through the use of an equation. The boy tells the story to his teacher and asks her why does his dad insist on the equation. The participants were asked to explain how they would answer the boy, if they were his teacher. By the end of the task round, we had 5 different types of solutions for a linear equation: i) algebraic one, ii) using diagrams (fraction bars), iii) trial and error, iv) simple false position, and v) another one by trial presented by the boy (i.e., 3 solutions by trial). We invited the participants to compare these solutions.

During the whole first task round, we tried to promote commognitive conflicts, since the discourse of the historical Egyptian source, the discourse of the solutions presented in the mathtasks are modelled by different metarules [START_REF] Sfard | Thinking as Communicating: Human development, the growth of discourses, and mathematizing[END_REF][START_REF] Kjeldsen | Bridging history of the concept of a function with learning of mathematics: Students' metadiscursive rules, concept formation and historical awareness[END_REF] and are (supposed to be) different from the teachers own's discourses. This strategy aimed to stimulate the participants to reflect about the way they deal with their students' different solutions and reasonings.

The first task round: what happened

The first task round took a bit more than two sessions, guided by the first two authors, and with the participation of 12 teachers (four of which attended all of the three meetings). During the sessions, we repeatedly asked the participants about how they deal with different reasonings and solutions brought by their students. Almost all participants have long-term experience in teaching mathematics at public and private schools. The sessions were audio and video recorded. There were separated audio recordings for each small group of 3 or 4 teachers, that discussed each task. The individual written answers to the mathstasks and to the historical task were collected. After each session, the participants were asked to write individual reflective diaries, in which they recorded their impressions of the session. Those data will be analysed in a next stage of the doctoral research project.

Impressions from the first activity's round implementation

In general, the participants engaged with the tasks and discussions conducted in the first task round.

We could discuss the approaches to equations and algebra teaching they use in classroom. When asked about how they deal with unusual solutions brought by their students, they acknowledge that they recognize them, sometimes sharing them with the rest of the class. However, although we tried to relativize the supremacy of algebra over other approaches, their discourses suggest that their main goal is to develop students' algebraic reasoning and writing. That is, even though they recognized some value on other approaches, they kept pursuing their goal, namely to lead students to understand that algebra makes solutions faster. For example, when we discussed solutions by trial, they repeatedly highlighted that if the numbers were bigger, they would have spent much more time, whilst algebraic solutions take the same amount of time regardless of how big the numbers are. None of them seemed to consider that experimenting with some examples could lead to an understanding of the problem, in such a way that testing all numbers would not be needed. One of the reasons for the teachers keep pursuing the algebra in classrooms might be the important role played by the algebra in the school syllabus in Brazil.

One of the participants, who we will refer to by the pseudonym Ulisses (who were present in all sessions) caught our attention in two different ways. Firstly, he told us that he used Ancient Egyptian multiplication algorithm (which had been discussed in the task round) with his students. He reported that, as his students were facing difficulties with the standard multiplication algorithm, he decided to show the Ancient Egyptian method. The students understood it and claimed that it was easier. We remark that we did not make any suggestions for the participants to apply specific approaches in their classrooms, and Ulisses did so by his own initiative. This kind of attitude is relevant to our work, since this teacher allowed himself to experiment in his own classroom with historical practices discussed during the sessions. During the discussion about Problem 25's solution, Ulisses once more surprised us: he thought of non algebraic solutions. Even though he wrote down the equation, he tried out different values. He noted that 10+5=15 and 12+6=18. Looking to 10+5=15, he noticed that 1 more was necessary, so he split 1 in three pieces 1/3, 1/3 and 1/3, of which two should be with 10 and one with 5and he reached the solution. Arcavi and Isoda (2007, p. 125) posit that the intention to nurture attentive listening by creating a gap between our method and the Egyptian had a slightly counterproductive effect for some: the symbolic method is so powerful and efficient why bother to even consider a complicated alternative so difficult to understand.

Contrary to this, in our case, the teachers considered all methodsof simple false position and the ones that were used in the mathtaskscorrect and valid.

Final Remarks

The continuation of our field work included two more task rounds. The second one explores Euclid's approach to areas without numbers (quadratures) and the role of the Pythagorean Theorem's in Euclidian tradition. The aim is to problematize the teaching of areas, which is mainly based on formulas in Brazil. The third task round is designed around the development of the concept of function, and some of its historical definitions, aiming to foster the reflection of image of mathematics as an immutable science.

The episodes reported above indicate that our proposal combining mathtasks and historical tasks seems to be a promising way to problematize teachers' understandings of mathematical knowledge production, in particular their students' mathematical knowledge production. When data analysis from our field work is completed, we will have more evidence to deepen the understanding of our proposal's potential. Nevertheless, the episodes from the first task round indicate that at least this group have been receptive to the use of the history of mathematics. We highlight that Ulisses spontaneously tried the Egyptian multiplication out in one of his classrooms. This indicates that he problematized, at least to some extent, not only his own and his students' mathematical knowledge productions, but also his teaching practices.

Figure 1 :

 1 Figure 1: Structure of task rounds

Figure 2 :Figure 3 :

 23 Figure 2: Hieratic and Hieroglyphic solutions of Problem 25

In Brazil, legal certification to teach at school is granted by an undergraduate degree. So, this master program is a professional development and is also a way to increase public school teachers' income.

Brazilian currency.

Problem 24 is quite similar to Problem 25 and states "A quantity whose seventh is added to it becomes 19."

Acknowledgments

We acknowledge CAPES for the scholarship for a doctoral internship at Rutgers University under the supervision of Prof. Arthur Powell. We are grateful to Prof Arthur Powell for the partnership, especially in the design of the field work methodology and to Prof. Elena Nardi and Dr. Irene Biza for the opportunity of working together and exchanging experience through the MathTASK project.


Slim Mrabet
email: mrabet_slim@yahoo.fr

The development of Thales theorem throughout history

Keywords: Thales theorem, similar triangles, transformation

Thales theorem may have different functionalities when using distances, algebraic measurements, or vectors. In addition to that, the utilization of a figure formed of secant lines and parallels or a figure relating to two similar triangles.

The aim of this work is to categorize different formulations of Thales theorem and explain why in teaching we must know the appropriate mathematical environment related to each Thales Theorem statement. The analysis of many geometry books in history makes it possible to distinguish two points of view according to different forms, demonstrations and applications of this concept.

The Euclidean point of view

The general statement of Thales theorem shows us the idea to move from one triangle to another, moreover, the link with similar triangles (immediately following it and generally with similar figures) is a characteristic of this point of view.

Proposal 2 of Book VI states that: "If a straight line be drawn parallel to one on the sides of a triangle, it will cut the sides of the triangle proportionally; and, if the sides of a triangle be cut proportionally, the line joining the points of section will be parallel to the remaining side of the triangle". (Heath, 1956).

The demonstration is based on the surfaces method; using equalities of triangles cases and making cuts and re-compositions in order to compare surfaces.

The point of view of transformations

This point of view is characterized by the disappearance of the link between Thales theorem and similar triangles, and, a passage from one line to another by projection appears in a figure of type: "parallels and secants".

In the seventeenth century, Euclidean treaty did not seem to satisfy some researchers. In fact, the latter prefer not to prove a result on lines using surfaces. In Thales theorem proof, Arnold rejects the detour by the surfaces made by Euclid.

For Arnold (1667), the statement of Thales theorem uses several parallels cut by secants: "If several lines, being differently inclined in the same parallel space, are all cut by parallel lines to this space, they are cut proportionally" When applying the Thales theorem, we find essentially the search of 4 th proportional, and the division of a segment with a given ratio.

This type of statement is also found in [START_REF] Hadamard | Leçons de géométrie élémentaires[END_REF] where the study of figures marks the appearance of transformations that coexist with traditional objects of geometry.

The homothety, introduced with distances, consolidates for Hadamard dynamic aspect of geometric figures. It also provides the possibility to use similarity of polygons by decomposing them into triangles.

Note that starting from the 20th century, new conceptions of geometry appeared and benefit from linear algebra. The traditional methods of Euclid and his successors were eventually set aside.

With [START_REF] Choquet | L'enseignement de la géométrie[END_REF], [START_REF] Dieudonné | Algèbre linéaire et géométrie élémentaire[END_REF] and other renowned researchers, geometry is evolving in each century. For Choquet, Thales theorem allows proving the relation    (u + v) = u + v , in order to study the linearity of an oblique projection.

Conclusion

In Thales theorem proof, Euclid uses surfaces method to avoid problems of irrational numbers. In majority of perused publications, Thales theorem proofs often use commensurable segments, thus, the transition to immeasurable segments is admitted.

In teaching, it is important for instructors to conceive the real line without holes. We also think that the two points of view of Thales theorem have their efficacy and both should have enough time to reach the students. We recommend set each one of them in the appropriate mathematical environment that can result in a better understanding of geometric themes introduced in class.

Figure 1 :

 1 Figure 1 : figure of Euclid


Marianne Thomsen

Inge Marie Olsen

Original

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Original sources, ICT and mathemacy

Marianne Thomsen 1 and Inge Marie Olsen 2 1 PhD student, Aarhus University & University College Copenhagen, Denmark; mtho@au.edu.dk 2 Mariendal Friskole, Copenhagen, Denmark; ingemarieolsen@hotmail.com Keywords: ICT, original sources, hidden mathematics, mathemacy.

The focus of our master's thesis in Mathematics Education was on how 4 th grade students (10-11 years old) connect working with ICT and reading original mathematical sources in ways that support the development of a competency, mathemacy, "which may help students to reinterpret their reality and to pursue a different reality" (Skovsmose & Nielsen, 1996, p. 1263). Our study indicated that ICT and original sources complement each other in terms of the students' learning activities. The students' experimental activities in ICT supported their understanding of the original source. Their work with the original source supported a development of a mathematical consciousness and sharpened their awareness of hidden mathematics [START_REF] Jankvist | The hidden mathematics of the mars exploration rover mission[END_REF] and black box situations [START_REF] Buchberger | Computer algebra: The end of mathematics[END_REF], which means that the students didn't just accept the results of a computerthey also questioned the mathematics behind when using ICT. Furthermore, the students' awareness of differences between working with contemporary ICT and original sources seemed to increase a balanced and critical stance towards mathematics as a universally given thingmaking them see both mathematics and ICT as something that develops and changes over time.

Our research design was inspired by [START_REF] Skovsmose | Kritisk forskning -paedagogisk udforskning[END_REF] description of critical research based on three situations: The current, the imagined and the arranged, as well as three processes: Pedagogical imagination, pedagogical experimentation and explorative analysis. The three situations and processes were used in our planning, implementation and analysis of the 3 x 90 minutes teaching session we conducted. The students were asked to carry out a final individual evaluation. We performed three types of data collection: 1) Focus group interviews prior to the first teaching session, 2) Teaching session observations, and 3) students' products from each teaching session. Data category 2 and 3 we regarded as pedagogical experimentation data helping us describe the current situation. We used this situation and our pedagogical imagination to prepare the next imagined situation. Based on the collected data we performed an explorative analysis and planned the next arranged situation and so on. The first current situation was based on data type 1. In our planning and analyses of the learning activities we were among other inspired by [START_REF] Jankvist | A categorization of the 'whys' and 'hows' of using history in mathematics education[END_REF] framework concerning the use of history in mathematical education as a goal (meta-issues) or as a tool (in-issues). Our study focused on history both as a goal and as a tool. We extended Jankvist's categorization also to deal with the students' work with ICT. The mathematical content was kept relatively simple letting the students work with only Euclid's Proposition 1, book I, To construct an equilateral triangle on a given finite straight line and equilateral triangles in GeoGebra. For the meta-issues we focused especially on hidden mathematics in software [START_REF] Jankvist | The hidden mathematics of the mars exploration rover mission[END_REF] and the fact that mathematics develops over time [START_REF] Jankvist | A categorization of the 'whys' and 'hows' of using history in mathematics education[END_REF]. Inspired by [START_REF] Jankvist | New avenues for history in mathematics educationmathematical competencies and anchoring[END_REF] and [START_REF] Kjeldsen | Beyond motivation: history as a method for learning metadiscursive rules in mathematics[END_REF], we used [START_REF] Sfard | Thinking as communicating[END_REF] theory of discourse and commognitive conflicts. We considered Euclid's proposition 1, book I, and GeoGebra as two different discourses. Within these two discourses we saw meta-issues and in-issues as two different discourses. During the teaching sessions, we made a common 'discourse-poster' on which we gathered the students' statements they agreed to during the classroom discussions. The students worked with the construction part and the proof part in relation to Euclid's Proposition 1, book I, and equilateral triangles in GeoGebra. This gave rise to common discussions about the difference between proof; in terms of Euclid's proposition and in terms of working with GeoGebra. One student for example compared Euclid's proof to a game; he found one circle, then the next and so on. The students were asked to construct as many equilateral triangles as possible in GeoGebra and formulate their own proofs for these. The outcome of this assignment was what they called three different types of 'proofs' for equilateral triangles: Measuring, the computer outcome as a proof in itself and the Euclidian way. There were different opinions among the students about which type of 'proof' they found most credible, which we regarded as an in-issue discussion; understandings of different kinds of proofs within the two discourses were to some extent generating commognitive conflicts. Furthermore, the students discussed whether the computer always tells the truth and if mathematic will change in the future as wella meta-issue discussion based on their experiences by working within the two different discourses. In addition to this, we arranged special learning activities, which focused on the meta-issues, for example asking students to write and discuss pros and cons of using Euclid's proposition and the use of GeoGebra in mathematical education. Several students found that working with both Euclid's proposition and GeoGebra was a good idea and had increased their learning. The students became aware of that computer outcome may contain a lot of hidden mathematics. Hopefully, this will support their reflexive use of ICT and also later in life qualify their critical stance towards the outcome of mathematical modelling.