Vincent Alberge
email: valberge@fordham.edu

Melkana Brakalova- And

Athanase Papadopoulos

TEICHM ÜLLER'S WORK ON THE TYPE PROBLEM

Keywords: Mathematics Subject Classification. 20F20, 28A75, 30F60, 32G15, 30C62, 30C75, 30C70, 30D30, 30D35 type problem, line complex, Riemann surface, quasiconformal mapping, quasiconformal extension, parabolic surface, hyperbolic surface, branched covering of the sphere, measure of ramification

Abstract. The type problem is the problem of deciding, for a simply connected Riemann surface, whether it is conformally equivalent to the complex plane or to the unit dic in the complex plane.

We report on Teichmüller's results on the type problem from his two papers Eine Anwendung quasikonformer Abbildungen auf das Typenproblem (An application of quasiconformal mappings to the type problem) (1937) and Untersuchungen über konforme und quasikonforme Abbildungen (Investigations on conformal and quasiconformal mappings) (1938). They concern simply connected Riemann surfaces defined as branched covers of the sphere. At the same time, we review the theory of line complexes, a combinatorial device used by Teichmüller and others to encode branched coverings of the sphere.

In the first paper, Teichmüller proves that any two simply connected Riemann surfaces which are branched coverings of the Riemann sphere with finitely many branch values and which have the same line complex are quasiconformally equivalent. For this purpose, he introduces a technique for piecing together quasiconformal mappings. He also obtains a result on the extension of smooth orientation-preserving diffeomorphisms of the circle to quasiconformal mappings of the disc which are conformal at the boundary.

In the second paper, using line complexes, Teichmüller gives a type criterion for a simply-connected surface which is a branched covering of the sphere, in terms of an adequately defined measure of ramification, defined by a limiting process. The result says that if the surface is "sufficiently ramified" (in a sense to be made precise), then it is hyperbolic. In the same paper, Teichmüller answers by the negative a conjecture made by Nevanlinna which states a criterion for parabolicity in terms of the value of a (different) measure of ramification, defined by a limiting process. Teichmüller's results in his first paper are used in the proof of the results of the second one.

The final version of this paper will appear in Vol. VII of the Handbook of Teichmüller theory (European Mathematical Society Publishing House, 2020).

Introduction: The type problem

By the uniformization theorem, every simply connected Riemann surface S is either conformally equivalent to the unit disc, or to the complex plane, or to the Riemann sphere. In the first case, the surface is said to be of hyperbolic type, in the second, of parabolic type, and in the third, of elliptic type. The last case is distinguished from the two others by topology: if S is contractible, it is not homeomorphic to the Riemann sphere. How can we distinguish between the other two cases? In other words, suppose that a contractible Riemann surface S is constructed by some process, for example, by assembling pieces which are themselves simply connected surfaces, or by analytic continuation, or-as in the papers considered-as an infinite branched covering of the sphere with information on the ramification index at the branching points. How can we decide if the surface is of hyperbolic or parabolic type? This is the so-called type problem.

Ahlfors writes in his comments to his Collected works [4, vol. 1, p. 84], that the type problem was formulated for the first time by Andreas Speiser 1 in his paper Probleme aus dem Gebiet der ganzen transzendenten Funktionen (Problems related to entire transcendental functions) [35] (1929). He stresses on the importance of this problem in his paper Quelques propriétés des surfaces de Riemann correspondant aux fonctions méromorphes (Some properties of Riemann surfaces corresponding to meromorphic functions) [START_REF] Ahlfors | Quelques propriétés des surfaces de Riemann correspondant aux fonctions méromorphes[END_REF] (1932), where he writes: "This problem is, or ought to be, the central problem in the theory of functions. It is evident that its complete solution would give us, at the same time, all the theorems which have a purely qualitative character on meromorphic functions."

Nevanlinna, in the introduction of his book Analytic functions (1953), writes (p. 1 of the English edition [26]): "[. . . ] Value distribution theory is thus integrated into the general theory of conformal mappings. From this point of view the central problem of the former theory is the type problem, an interesting and complicated question, left open by the classical uniformization theory."

Teichmüller addressed the type problem in several papers. We shall report here on his work on this subject in his two papers Eine Anwendung quasikonformer Abbildungen auf das Typenproblem (An application of quasiconformal mappings to the type problem) [START_REF] Teichmüller | Eine Anwendung quasikonformer Abbildungen auf das Typenproblem[END_REF], published in 1937, and Untersuchungen über konforme und quasikonforme Abbildungen (Investigations on conformal and quasiconformal mappings) [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF], published in 1938.

The Riemann surfaces that are considered by Teichmüller in these two papers are coverings of the sphere of the sphere branched over a finite number of values. Nevanlinna, in his papers [START_REF] Nevanlinna | Über die Riemannsche Fläche einer analytischen Funktion[END_REF][START_REF] Nevanlinna | Über Riemannsche Fläche mit endlich vielen Windunspunkten[END_REF] and in his book [26] (in particular in §XI.2 and §XII.1), studies a particular class of such surfaces. A particularly important class of examples is constituted by surfaces associated with automorphic functions, but there are many others. As a matter of fact, the idea of studying a Riemann surface which is a branched cover of the sphere and the problem of reconstructing it from the combinatorial information at the branch points and branch values originates with Riemann. The latter introduced Riemann surfaces in his doctoral dissertation [START_REF] Riemann | Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse[END_REF] and described, for the first time, the domain of a meromorphic function to be a branched covering of the sphere. In this setting, he talked about surfaces "spread over the sphere" ( Überlagerungsfläche). The problem of reconstructing the Riemann surface from combinatorial information at the branch points is a form of the so-called Riemann existence problem [START_REF] Riemann | Theorie der Abel'schen Functionen[END_REF]; see also Stoïlov's work [START_REF] Stoïlov | Leçons sur les principes topologiques de la théorie des fonctions analytiques : Professées à la Sorbonne et à l'université de Cernauti. 2e édition[END_REF] on the so-called Brouwer problem, asking for a topological characterization of an analytic function. All this includes the kind of problems considered in Teichmüller's papers in a broader contex which can be traced back to the origin of Riemann surface theory.

We shall also review some fundamental tools that appear in Teichmüller's papers [START_REF] Teichmüller | Eine Anwendung quasikonformer Abbildungen auf das Typenproblem[END_REF] and [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF], in particular the notion of line complex that was extensively used by Nevanlinna and others, which is a topological/combinatorial tool that describes a type of branched covering. Independently of the work reviewed here, the ideas around the theory of line complexes associated to Riemann surfaces should be interesting to low-dimensional geometers and topologists.

Teichmüller, in his work on the type problem, also relied on quasiconformal mappings. As a matter of fact, the paper Eine Anwendung quasikonformer Abbildungen auf das Typenproblem [START_REF] Teichmüller | Eine Anwendung quasikonformer Abbildungen auf das Typenproblem[END_REF], whose subject is the type problem, is the first one written by Teichmüller in which he uses these mappings. Teichmüller made extensive use of quasiconformal mappings in his approach to the type problem as well as in other works he did on complex analysis, namely, his papers on Riemann's moduli problem [START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF] [46] [START_REF] Teichmüller | Vollständige Lösung einer Extremalaufgabe der quasikonformen Abbildung[END_REF], on the Bieberbach coefficient problem [START_REF] Teichmüller | Ungleichungen zwischen den Koeffizienten schlichter Funktionen[END_REF] and on value distribution theory [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF] [START_REF] Teichmüller | Ungleichungen zwischen den Koeffizienten schlichter Funktionen[END_REF].

Line complexes

We recall the definition of a line complex associated with a branched covering M of the Riemann sphere. This is a graph embedded in M which carries the combinatorics of this covering. We shall follow the exposition in Nevanlinna's book [26], which is also the one Teichmüller uses in the two papers [START_REF] Teichmüller | Eine Anwendung quasikonformer Abbildungen auf das Typenproblem[END_REF] and [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF].

We assume that M is a simply connected Riemann surface which is a branched covering of the Riemann sphere S 2 = C ∪ {∞}, branched over finitely many points a 1 , . . . , a q ∈ S 2 (q ≥ 2) called the branching values. Such a point is either of algebraic type (that is, the corresponding branching degree is finite) or of logarithmic type (that is, the corresponding branching degree is infinite). Nevanlinna considers the class of such surfaces in [26, Chapter XI, §2 and §3] and he denotes a surface belonging to this class by F q , or by F (a 1 , • • • , a q ) if the branching values are prescribed. We shall keep using Teichmüller's notation M. One example of such a surface is the universal covering of the sphere punctured at a finite set of points a 1 , . . . , a q , q ≥ 2 (see [26, §I.3] for further details). Nevanlinna says that this surface is distinguished among the others as being "as ramified as possible" [26, p. 290].

To define the line complex associated to M, we start by drawing a simple closed curve γ passing through the points a 1 , . . . , a q in the given cyclic order (chosen arbitrarily). This curve decomposes the sphere into two simply connected regions I and A that have a natural polygonal structures with vertices a 1 , . . . , a q and sides (a 1 a 2 ), (a 2 a 3 ), . . . (a q a 1 ). The pre-image of γ by the covering map M → S 2 is a graph G which divides M into a finite or infinite number of pieces that are equipped with a natural polygonal structure induced from either I or A. These pieces are called "half sheets," and are labeled by I, A.

The line complex is a graph embedded in the surface M which encodes the way the half sheets are glued together. We recall now more precisely the definition.

If ψ : M → S 2 denotes the branched covering map, then this map restricted to the set M \ ψ -1 ({a 1 , . . . , a q }) is an unbranched covering map onto its image S 2 \ {a 1 , . . . , a q }. A point lying above a branch value a i (i = 1, . . . , q) (or, equivalently, a vertex of a half sheet in M, allowing also a vertex to be at infinity), may be of three kinds:

(1) Infinite order: such a vertex is not in M but lies at infinity.

(2) Finite order: there are 2m half sheets glued cyclically at such a point; in this case the point is said to be of order m -1. (3) An unbranched point, called also "nonliteral". These are points of order zero.

We construct a dual graph to the polygonal curve γ in the sphere S 2 by choosing two points P 1 and P 2 in the interior of the polygons I and A respectively and joining, for each edge (a i a i+1 ) of γ, P 1 and P 2 by a simple arc that crosses γ at a single point in the interior of that edge. This defines the dual graph of γ. Its lift by ψ -1 , as a planar graph embedded in M, is the line complex Γ of the covering.

The line complex Γ might be finite or infinite (in the sense that it may have a finite or infinite number of edges) depending on whether the Riemann surface M is a finite or infinite branched covering of the sphere. As a graph, it is homogeneous of degree q, and it is bipartite: choosing a coloring, say black and white, for the two vertices P 1 or P 2 , the vertices of Γ can also be colored black and white, depending on whether they are lifts of P 1 and P 2 , and such that any edge of Γ joins a white vertex to a black vertex.

The line complex Γ decomposes the surface M into pieces called "elementary regions" (the complementary components of the graph imbedded in that surface). Each elementary region is a polygon. Such a polygon corresponds by the covering projection ψ to a unique branch value a i on the sphere. Corresponding to the above classification of the branch points, the polygons are of three types:

(1) A polygon with an infinite number of sides. It is unbounded in M and it corresponds to a logarithmic ramification point.

(2) A polygon with an even number 2m of sides, containing a ramification point of finite order m -1. (3) A bigon (containing an unramified point over an a i ). Line complexes were introduced by Nevanlinna in [START_REF] Nevanlinna | Über die Riemannsche Fläche einer analytischen Funktion[END_REF] and Elfving in [START_REF] Elfving | Uber eine Klasse von Riemannschen Flächen und ihre Uniformisierung[END_REF]. They were used as an important combinatorial tool by Nevanlinna, Ahlfors, Speiser, Elflving, Ullrich, Teichmüller and other mathematicians in their investigation of the type problem. In this setting, the type problem is reduced to that of recovering the type of a Riemann surface from the properties of its line complex.

As Nevanlinna and Ahlfors put it several times, the idea of encoding a branched covering of the sphere by a graph is contained in the paper [START_REF] Speiser | Über Riemannsche Flächen[END_REF] by Speiser. 2 In fact, the latter introduced an object that was called later Speiser tree, as a topological tool to study the type problem. The definition of a Speiser tree is close to that of a line complex, but we do not need to recall here the difference between the two notions.

The branched covering M of the sphere is uniquely determined by the points a 1 , . . . , a q , the simple closed curve γ joining them and the line complex Γ. In the papers [START_REF] Elfving | Uber eine Klasse von Riemannschen Flächen und ihre Uniformisierung[END_REF] by Elfving and [24] by Nevanlinna, Riemann surfaces are constructed from a given line complex satisfying certain conditions. There are sections on line complexes in the books by Nevanlinna [26] and [27], by Sario and Nakai [START_REF] Sario | Classification Theory of Riemann Surfaces[END_REF], by Goldberg and Ostrovskii [START_REF] Goldberg | Value Distribution of Meromorphic Functions[END_REF], and in other works.

Teichmüller's paper Eine Anwendung quasikonformer Abbildungen auf das Typenproblem

Teichmüller's motivation in his paper Eine Anwendung quasikonformer Abbildungen auf das Typenproblem [START_REF] Teichmüller | Eine Anwendung quasikonformer Abbildungen auf das Typenproblem[END_REF] is the study of the type problem based on the properties of line complexes. He asks: "How can one determine, from the given line complex, if the corresponding surface can be mapped one-to-one and conformally onto the whole plane, the punctured plane or the unit disc?" He notes that "one is still very far away from sufficient and necessary criteria."

Let us start by making a list of the main ideas and results contained in that paper:

(1) The question of determining the type of a Riemann surface from the properties of its line complex. (2) The formula

D = |K| + K 2 -1 with K = 1 2 u 2 x + u 2 y + v 2 x + v 2 y u x v y -v x u y .
for the dilatation quotient at a point of a map x + iy → u + iv with continuous partial derivatives, and the definition of a 2 In the paper [START_REF] Ahlfors | Riemann surfaces and small point sets[END_REF], Ahlfors writes: "Around 1930 Speiser had devised a scheme to describe some fairly simple Riemann surfaces by means of a graph and had written about it in his semiphilosophical style".

quasiconformal mapping as a mapping with bounded dilatation quotient.

(3) The fact that the type of a simply connected Riemann surface is a quasiconformal invariant. (4) The fact that the unit disc and the complex plane are not quasiconformally equivalent. [START_REF] Ahlfors | Riemann surfaces and small point sets[END_REF] The fact that any two simply connected Riemann surfaces defined as branched coverings of the Riemann sphere with finitely many branch values and which have the same line complex are quasiconformally equivalent. ( 6) Techniques for piecing together quasiconformal mappings. [START_REF] Beniamini | A negative answer to Nevanlinna's type question and a parabolic surface with a lot of negative curvature[END_REF] The extension of an orientation-preserving diffeomorphism of the circle of class C 1 to a quasiconformal mapping of the disc which is conformal at the boundary.

These items are inter-related. For instance, (3) is equivalent to (4), ( 7) is used in the proof of (6), etc.

We now discuss in more detail the content of the paper [START_REF] Teichmüller | Eine Anwendung quasikonformer Abbildungen auf das Typenproblem[END_REF].

Teichmüller proves the following:

Theorem 3.1. Any two Riemann surfaces with the same line complex have the same type.

In order to prove this result, Teichmüller shows the following key results which we state as propositions: Proposition 3.2. Any two branched coverings of the sphere with equal line complexes can be mapped quasiconformally onto each other. The latter proposition is a direct consequence of the following result: Proposition 3.4. The complex plane and the unit disc cannot be conformally mapped onto each other.

After stating his results, Teichmüller, in §2 of his paper, defines the notion of "dilatation quotient" of a mapping at a point where the mapping is of class C 1 .

We recall that the differential of a C 1 mapping takes an infinitesimal ellipse centered at a point to an infinitesimal ellipse centered at the image. The dilatation quotient of a C 1 mapping f is a function that assigns to each point of the domain of f the ratio of the major axis to the minor axis of an ellipse that is the image of a circle centered at the origin by the differential of f . This dilatation quotient is also known as the distortion of the Tissot indicatrix ; see Grötzsch's paper [START_REF] Grötzsch | Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche[END_REF] where the author uses this expression, and the review [START_REF] Papadopoulos | Quasiconformal mappings in the work of Nicolas Auguste Tissot[END_REF] on Tissot's work in this volume. There is also a relation with the characteristic functions introduced by Lavrentieff in [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF] and [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF]. Teichmüller derives formula (2) above, which allows one to compute the dilatation quotient in terms of the partial derivatives of the mapping.

In §3, Teichmüller first defines a quasiconformal mapping as a continuous one-to-one and sense-preserving mapping between two domains of the plane that is differentiable except at isolated points and whose dilatation quotient is bounded. This definition is close to the one given by Grötzsch in [START_REF] Grötzsch | Über die Verzerrung bei schlichten nicht konformen Abbildungen und über eine damit zussamenhangende Erweiterung des Pikardschen Satzes[END_REF]. The differentiablity conditions in the definition of a quasiconformal mapping were relaxed later; cf. for instance Lehto's historical survey on quasiconformal mappings [START_REF] Lehto | A historical survey of quasiconformal mappings[END_REF].

Teichmüller notes that the definition of a quasiconformal mapping between domains of the plane can be transferred to the setting of mappings between Riemann surfaces. He then gives conditions for piecing together continuously differentiable and quasiconformal mappings defined on two domains S 1 and S 2 separated by a curve C. He concludes this section by solving what he describes as "an important special mapping problem," namely, the problem of extending an arbitrary orientation-preserving C 1 diffeomorphism of the unit circle S 1 to a quasiconformal mapping of the unit disc D that is "conformal at the boundary." Here, a differentiable mapping of the unit disc is said to be conformal at the boundary if its dilatation quotient on sequences of points approaching the boundary approaches 1.

Remarks 3.5. 1. The extension that Teichmüller constructs is (up to a clockwise rotation by π 4 ) the one that appears in Ibragimov's paper [START_REF] Ibragimov | Quasi-isometric extension of quasisymmetric mappings of the real line compatible with composition[END_REF], in which the latter says that such an extension was known to Ahlfors and Beurling (without any further reference). Presumably, Ibragimov was not aware of Teichmüller's result.

2. A result of Ahlfors and Beurling says that a circle homeomorphism can be extended quasiconformally to the unit disc if and only if it is quasisymmetric (see [START_REF] Beurling | The boundary correspondence under quasiconformal mappings[END_REF]). Today, two well-known quasiconformal extensions of quasi-symmetric homeomorphisms of the circle are the Beurling-Ahlfors and the Douady-Earle extensions, introduced in the papers [START_REF] Beurling | The boundary correspondence under quasiconformal mappings[END_REF] and [START_REF] Douady | Conformally natural extension of homeomorphisms of the circle[END_REF] respectively. The two extensions are different from the one described by Teichmüller.

3. Teichmüller's notion of quasiconformal mappings of the disc that are "conformal at the boundary" is used today in the definition of "asymptotically conformal" quasiconformal mappings of the disc, a class of mappings that are at the basis of the notion of asymptotic Teichmüller space of the disc introduced by Gardiner and Sullivan in their paper [START_REF] Gardiner | Symmetric structure on a closed curve[END_REF].

In §4, Teichmüller proves that two surfaces having the same line complexes can be mapped quasiconformally onto each other (Proposition 3.2 above). With the notation introduced for line complexes in §2, after mapping the two discs A and I quasiconformally onto the inside and the outside of the unit disc respectively, he uses the extension he introduced in the previous section in such a way that the mapping obtained is conformal on the simple closed curve γ (the common boundary of the discs A and I), and defining therefore a quasiconformal mapping between the z-plane and the w-plane. He performs the same operation with another Riemann surface M ′ that has the same line complex as M. In this way, he gets a quasiconformal mapping from a z ′ -plane onto the w-plane and then a quasiconformal mapping from the z-plane onto the z ′ -plane. Because M and M ′ share the same line complex, the gluing specification between corresponding half sheets is the same. Teichmüller eventually gets a quasiconformal mapping between M and M ′ .

In §5, he proves that the unit disc and the complex plane cannot be quasiconformally mapped onto each other (Proposition 3.4 above). The proof is by contradiction and it is based on the so-called lengtharea method, a method to which Teichmüller referred in his later papers as the Grötzsch-Ahlfors method (see for instance [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF][START_REF] Teichmüller | Extremale quasikonforme Abbildungen und quadratische Differentiale[END_REF][START_REF] Teichmüller | Über Extremalprobleme der konformen Geometrie[END_REF][START_REF] Teichmüller | Vollständige Lösung einer Extremalaufgabe der quasikonformen Abbildung[END_REF]). More precisely, Teichmüller reasons by contradiction. He assumes that there exists a quasiconformal mapping from the complex plane to the unit disc. Such a mapping sends, for any 0 < r 1 < r 2 , the annulus {z | r 1 < |z| < r 2 } onto a doubly-connected subdomain of the unit disc. The image being bounded, it has a bounded conformal modulus and therefore the modulus of {z | r 1 < |z| < r 2 } is also bounded. This is obviously false since r 2 can be chosen arbitrarily large.

In an amendment made during the page proofs of his paper and written at the end of §5, Teichmüller notes that the final part of the paper [START_REF] Grötzsch | Über die Verzerrung bei schlichten nicht konformen Abbildungen und über eine damit zussamenhangende Erweiterung des Pikardschen Satzes[END_REF] by Grötzsch published in 1928 is closely related to his own proof of Proposition 3.3 (information he obtained from H. Wittich). 3 In that paper, Grötzsch uses this method in order to extend the small and the big Picard theorems to quasiconformal (and not only conformal) mappings. Lavrentieff uses the same idea in his 1935 paper [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF] which we mention below. Conceivably, Teichmüller was not aware of Lavrentieff's paper. 4 In the last section, Teichmüller provides an example to briefly show that for a simply connected Riemann surface M defined as a branched covering of the sphere with branch values a 1 , • • • a q , the position of these points and the simple closed curve γ joining them that is used in the construction of the line complex, although it has no influence on the question of deciding the type of the surface, has an influence on the value distribution of the uniformizing mapping, namely, the biholomorphic mapping that sends M onto the disc or to the complex plane. He deals with the explicit example of a branched covering with four branch values and with a line complex determined by the intersection of the real line with infinitely many circles. He gives explicitly the Nevanlinna characteristic of the associated meromorphic mapping and shows that 3 A translation of Grötzsch's paper is provided in the present volume. 4 Teichmüller was aware of Lavrentieff's paper when he wrote the later paper [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF]; see the footnote in §6.4 of [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF].

the coefficients of this characteristic depend on the choice of γ. He declares that the proof is not difficult but will be carried out in a different context. The same example is further considered in his 1944 paper [START_REF] Teichmüller | Einfache Beispiele zur Wertverteilungslehre[END_REF].

Teichmüller was not the first to be interested in the application of quasiconformal mappings to the type problem, and we refer here again to the 1935 work of Lavrentieff in [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF] (see the English translation in the present volume; cf. also [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF] for a summary of the results and the paper [START_REF] Alberge | A commentary on Lavrentieff's paper Sur une classe de représentations continues[END_REF] for a commentary also in the present volume). Lavrentieff introduced the notion of fonction presque analytique (almost analytic function), a class of mappings that are more general than the ones introduced earlier by Grötzsch, and whose dilatation quotient is not necessary bounded. He gives, for a particular class of Riemann surfaces defined as the graph of a function of two variables, a sufficient condition on the dilatation quotient that makes the surface of hyperbolic type (see [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF]Théorème 8]). The same year Teichmüller published his paper [START_REF] Teichmüller | Eine Anwendung quasikonformer Abbildungen auf das Typenproblem[END_REF] (1937), Kakutani published a paper [START_REF] Kakutani | Applications of the theory of pseudo-regular functions to the typeproblem of Riemann surfaces[END_REF] in which he also studies the type problem using (another) notion of quasiconformal mappings. See also Kobayashi's paper [START_REF] Kobayashi | On the Kakutani's theory of Riemann surfaces[END_REF].

In his paper [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF] which we consider in §5, Teichmüller gives a criterion to find the type of Riemann surfaces in a class already considered in his paper [START_REF] Teichmüller | Eine Anwendung quasikonformer Abbildungen auf das Typenproblem[END_REF], and as a consequence he constructs an example of a hyperbolic Riemann surface with a so-called "mean ramification" equal to 2. We review this in the next section.

Nevanlinna's conjecture

The notation is the one introduced in §2. Nevanlinna's conjecture is based on the fact that the type of the simply connected surface M may be deduced from information on the measure of the amount of ramification of its associated line complex. Roughly speaking, if the amount of ramification is small (in some sense to be defined), the surface is of parabolic type, and if it is large, it is of hyperbolic type. The motivation is that a large degree of ramification will put a large angle structure at the vertices of the graph G. This can be made precise by introducing a notion of combinatorial curvature, and it is the object of these investigations. Nevanlinna writes that "it is natural to imagine the existence of a critical degree of ramification that separates the more weakly branched parabolic surfaces from the more strongly branched hyperbolic surfaces" [26, p. 308]. To make precise this observation, he introduces a notion of curvature of a line complex, which he calls the mean excess. His conjecture says that the surface M is parabolic or hyperbolic depending on whether the mean excess of its line complex is zero or negative respectively. Teichmülller in his paper [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF], disproves the conjecture by exhibiting a hyperbolic simply connected Riemann surface branched over the sphere whose mean excess of the corresponding line complex is equal to zero.

Nevanlinna's study of this topic is restricted to the class of surfaces F q that we defined earlier (simply connected surfaces that are branched covers of the sphere and whose branch points project to a finite number of points a 1 , . . . , a q ). We quote him from his book [26, p. 312]:

[. . . ] Relative to transcendental surfaces Fq the following question arises: Is the surface parabolic or hyperbolic according to the angle geometry of the surface Fq is "euclidean" or "Lobachevskyan," i.e. according as the mean excess E = 2 -V is zero or negative? This question was answered (1938) by Teichmüller [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF] and, indeed, in the negative.

In the rest of this section, we explain Nevanlinna's conjecture. We use the notation for line complexes introduced in §2. The vertices of the complex Γ correspond to the half sheets in M and the faces of Γ, which are all homeomorphic to discs, are called elementary polygons. The number of sides of an elementary polygon may be infinite, but if it is finite, then it is even. Indeed, an elementary polygon covers a cell in the Riemann sphere whose boundary consists of two edges. Therefore, each face has 2m edges, with m = 1, 2, . . . , ∞. Each elementary polygon with 2m sides is associated with a branch point of order m -1.

In order to give an idea of the ramification measure that he introduces, Nevanlinna first considers the case of a closed surface which is a finite-sheeted covering of the sphere. In this case, the ramification measure is the sum of the orders of the branch points divided by the number of sheets of the covering. The calculation in this special case is made in terms of the associated line complex.

The goal is to assign a ramification measure to each vertex of the graph. (Recall that the ramification points of the covering map ψ from M to the Riemann sphere are at the "centers" of the polygons that are the complementary components of the line complex.)

To each branch point of order m -1 corresponds an elementary polygon with 2m sides and 2m vertices. Twice the order of the branch point, that is, 2m-2, is distributed onto the 2m vertices of the polygon. In this way, each vertex P of Γ receives from the given elementary polygon a contribution of the line complex 2m-2 2m = 1 -1 m of the order of ramification. Therefore, the total ramification at a vertex P is equal to

V P = W (1 - 1 m ),
the sum, for each P , being taken over the elementary polygon W having P as a vertex. Since the number of vertices of Γ is equal to 2n (twice the number of sheets of the covering), the mean ramification of the surface, which is naturally defined as 1 2n

P V P , is equal to 2 - 2 n .
We now return to our simply connected infinitely-sheeted surface M, branched over q points, with associated line complex Γ.

As in the previous special case, to each vertex P of Γ is associated a ramification index V P coming from its adjacent elementary polygons. With the above notation, the index coming from one polygon is equal to 1 -1 m and since the number of edges of the elementary polygons may be infinite, we allow m to be infinite, in which case the ramification index coming from this region is 1.

Since P is a vertex of at least 2 and at most q adjacent elementary regions whose order m -1 is positive, we always have 1 ≤ V P ≤ q.

Since the surface M has infinitely many sheets, to define the average ramification of the associated line complex Γ, Nevanlinna uses what he calls a wreathlike exhaustion (see [26, §XII.1.3]) of this graph by an infinite sequence of subgraphs Γ ν (ν = 1, 2, . . .). This is done by starting with a vertex P 0 chosen arbitrarily, which we call the base vertex, then defining the complex Γ 1 as the subgraph of Γ obtained by adding to P 0 the vertices P 1 at distance one from this base vertex (first generation) together with the edges joining these new vertices to P 0 , then the complex Γ 2 as the one obtained by adding the vertices P 2 at distance one from Γ 1 (second generation) together with the edges joining these new vertices to Γ 1 , etc. Here, the distance between two vertices is defined as the minimal number of edges joining them.

For every ν = 1, 2, . . ., we let n ν be the number of vertices in the approximating graph Γ ν . The mean ramification of Γ ν is defined as

V ν = 1 n ν F ν V P .
The mean ramification of Γ (and of the surface M) is the limit

V = lim ν→∞ V ν ,
provided this limit exists. If this limit does not exist, Nevanlinna works with the lower limit lim V ν and the upper limit lim V ν .

There is a geometric way of calculating this measure of ramification, in relation with a combinatorial notion of curvature. We return to the graph G = ψ -1 (γ) in M, that is, the pre-image by the projection map of the simple closed curve γ in the Riemann sphere that joins the branchvalues a 1 , . . . , a q . We recall that G defines a combinatorial decomposition of M whose cells have q sides and whose vertices lie above the branching values a 1 , . . . a q . There is a natural combinatorial angle structure associated to this polygonal decomposition of M, where to each vertex of G which corresponds to a branch point of order m -1 one associates an angle measure equal to π/m. The excess of a polygon P with q sides is then defined as

E P = 1 m -q + 2
where the sum is taken over the angles of the polygon. This is justified as follows:

If the q-sided polygon were a Euclidean polygon, then its angle sum would be equal to (q -2)π. Suppose now that we are given a q-sided polygon which is equipped with a metric or more generally with a structure that assigns to it a reasonable angle structure at the vertices. Then, the "excess" of this polygon, which is a measure of its deviation from being Euclidean, is taken to be its angle sum divided by π minus the angle sum of a Euclidean polygon with the same number of sides divided by π. This is consistent with the formula we gave for E P .

The above formula for the ramification V P of a polygon P gives

V P = (1 - 1 m ) = q - 1 m .
Thus, we have

V P + E P = 2,
that is, the sum of the ramification and the excess of a fundamental polygon P is equal to 2.

To get a more precise idea of the geometric meaning of ramification, we consider the case of a regularly ramified surface, where V = V P for every fundamental polygon P . There are three cases:

(1) Elliptic case: V < 2. One can take the fundamental polygons on M to be spherical polygons with geodesic sides and angle excess πE (E = E P being the excess defined above). (2) Parabolic case: V = 2, E = 0. The fundamental polygons can be taken as Euclidean polygons, with zero excess. (3) Hyperbolic case: V > 2, E < 0. The fundamental polygons can be taken as hyperbolic polygone, with angle deficit equal to π|E|.

This motivates the following conjecture which Nevanlinna made in the general case of a non-necessarily regular surface [26, p.312]: The surface is parabolic (respectively hyperbolic) if the mean excess E = 2 -V is zero (respectively negative).

Teichmüller, in §7 of the paper [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF] disproves the conjecture by exhibiting a hyperbolic simply connected surface S branched over the sphere and whose mean excess is zero. We review Teichmüller's work in the next section.

Let us note that in a more recent paper [START_REF] Beniamini | A negative answer to Nevanlinna's type question and a parabolic surface with a lot of negative curvature[END_REF], the authors disprove the second part of the conjecture by giving an example of a simply connected Riemann surface which is parabolic and which has negative mean excess.

5.

The type problem in Teichmüller's paper Untersuchungen über konforme und quasikonforme Abbildungen

In this section, we review the criterion for hyperbolicity of surfaces in a certain class given by Teichmüller in the last section of his paper Untersuchungen über konforme und quasikonforme Abbildungen. The section is titled A type criterion.

Teichmüller considers a class of surfaces M satisfying the following properties (we use the notation of the previous section):

(1) no branch point is algebraic;

(2) the line complex Γ associated to M does not have any infinite chain of edges with no branch points; (3) each vertex of Γ has either two or q edges starting from it. Teichmüller proves that such a surface is hyperbolic if and only if it is sufficiently ramified, and for this, introduces a notion of measure of ramification which is different from the one introduced by Nevanlinna ( §4 above). We recall the definition.

We take a vertex B 0 adjacent to q simple edges (by Conditions ( 2) and ( 3), such a vertex exists). From B 0 , we follow, from each edge adjacent to it, a chain of edges until we arrive at a branch vertex. We define in this way q vertices that are branch points, and we call this set of vertices B 1 . By Condition (3), each vertex in B 1 has q -1 outgoing branches. Continuing in the same manner, we obtain q(q -1) branch vertices B 2 , then q(q-1) 2 branch vertices B 3 , etc. Thus, for each k ≥ 1, there are q(q -1) k-1 chains leading from the set of vertices B k-1 to the set of vertices B k . Condition (1) says that continuing in this way we traverse the entire line complex without repetitions.

Each chain of vertices has a well-defined length, and Teichmüller denotes by ϕ(k) the maximum of the lengths of all chains connecting an element in B k to an element in B k-1 . For each k ≥ 1, he sets

ψ(k) = Max κ=1,...,k ϕ(k).
The smaller ϕ(k) and ψ(κ) are, the more ramified is the surface M. Teichmüller proves the following:

Theorem 5.1. If the series ∞ k=1 ψ(k) k 2 converges, then M is of hyper- bolic type.
The proof uses an explicit conformal representation onto the unit disc and hyperbolic geometry. It is based on the construction of a quasiconformal mapping (with a non-necessarily bounded dilatation) from M onto the modular surface (the universal covering of the Riemann sphere punctured at the points a 1 , a 2 , . . . , a q ), which is known to be hyperbolic. In order to conclude with the invariance of the type, Teichmüller uses a result he previously obtained (see [41, §6.5]) and which is a generalisation of Proposition 3.4. The quasiconformal mapping from M onto the modular surface is constructed by first modifying the line complex associated to M and making it a Speiser tree, by replacing all chains without branching by bundles of edges arranged in a row. Teichmüller gives then an example of a surface M whose mean ramification V in the sense of Nevanlinna ( §2 above) is equal to 2, while ∞ k=1 ψ(k) k 2 converges. Thus, the surface is hyperbolic, which disproves Nevanlinna's conjecture.

In an addendum to his paper written on the page proofs, Teichmüller notes that Kakutani, in his paper [START_REF] Kakutani | Applications of the theory of pseudo-regular functions to the typeproblem of Riemann surfaces[END_REF] (1937), obtained a sufficient criterion for hyperbolicity that applies to a class of surfaces he considers in § 7.3 of his paper [START_REF] Teichmüller | Untersuchungen über konforme und quasikonforme Abbildung[END_REF]. He says that Kakutani's criterion does not contain his own criterion, but gives a better order of magnitude. He then writes about Kakutani: "If he had worked with unbounded dilatation coefficients and applied my § 6.5, he would have obtained a much better criterion for § 7.3 than the one developed here. His quasiconformal mapping is hence more suitable than mine; it is (for now) restricted to the special surfaces in §7.3."

Remarks 5.2. 1.-The results on the type problem obtained in Teichmüller's papers considered here concern surfaces which are branched coverings of the Riemann sphere with logarithmic branch points. There is a result of Ahlfors on the type problem for branched coverings with only algebraic branch points, see [START_REF] Ahlfors | Zur Bestimmung des Typus einer Riemannschen Fläche[END_REF] (1931) and results by Speiser [START_REF] Speiser | Über Riemannsche Flächen[END_REF][START_REF] Speiser | Über beschtänkte automorphe Funktionen[END_REF] and Ullrich [START_REF] Ullrich | Über ein Problem von Herrn Speiser[END_REF]. Ahlfors' first works on the type problem (see [START_REF] Ahlfors | Zur Bestimmung des Typus einer Riemannschen Fläche[END_REF] and [START_REF] Ahlfors | Sur le type d'une surface de Riemann[END_REF]), unlike the works of Laverentieff on the same problem, involve the so-called length-area method which is a common tool used by Grötzsch and himself, and later by Teichmüller, in relation to quasiconformal mappings.

2.-We already noted that Lavrentieff, before Teichmüller, used quasiconformal maps (in the version he called "almost analytic") in the study of the type problem (cf. [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF] and [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF]). His approach was Riemannian, unlike the combinatorial approach of Teichmüller. See the translation of Lavrentieff's paper [START_REF] Lavrentieff | Sur une classe de représentations continues[END_REF] in the present volume and the corresponding commentary [START_REF] Alberge | A commentary on Lavrentieff's paper Sur une classe de représentations continues[END_REF]).

4.-There is a large literature on the type problem. Kakutani made the relation with Brownian motion. In [START_REF] Kakutani | Two-dimensional Brownian motion and the type problem of Riemann surfaces[END_REF] (1945), he proved the following: Let S be a simply-connected open Riemann surface which is an infinite cover of the sphere. Let D be a simply connected subsurface of S which is bounded by a Jordan domain Γ. For a point ζ in S \ D, let u(ζ) be the probability that the Brownian motion on S starting on ζ enters into Γ without getting out of S -D before. Then, one of the following two cases holds:

(1) u(ζ) is < 1 everuwhere in S -D, and in this case S is hyperbolic (2) u(ζ) is identically equal to 1 on S -D, and in this case S is parabolic.

Doyle made a relation between the type problem, Brownian motion and the propagation of an electric current on the surface, see [START_REF] Doyle | Random walks and electric networks[END_REF]. In fact, this is a return to Riemann's point of view. Indeed, the latter relied, at several places in his work on Riemann surfaces, to arguments from electricity (see Riemann's doctoral dissertation [START_REF] Riemann | Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse[END_REF] and his paper on Abelian functions [START_REF] Riemann | Theorie der Abel'schen Functionen[END_REF]. The interested reader may also refer to the survey [START_REF] Papadopoulos | Physics in Riemann's mathematical papers[END_REF] on physics in Riemann's mathematical work).

Index

Ahlfors, Lars

Proposition 3 . 3 .

 33 The type of a simply connected open Riemann surface is invariant under quasiconformal mappings.

1

  Andreas Speiser (1885[START_REF] Nevanlinna | Analytic functions[END_REF] was a Swiss mathematician. He studied in Göttingen, first with Minkowski and then with Hilbert, who became his doctoral advisor after the death of Minkowski. He defended his PhD thesis, in Göttingen, in 1909 and his habilitation in 1911, in Strasbourg. Speiser worked in group theory, number theory and Riemann surfaces, but he is mostly known as the main editor of Euler's Opera Omnia. (He edited 11 volumes and collaborated to 26 others.)

Acknowledgements. The first and third authors acknowledge support from the U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 "RNMS: GEometric structures And Representation varieties" (the GEAR Network).


Seán Delaney
email: sean.delaney@mie.ie

Ann Marie Gurhy
email: annmarie.gurhy@mie.ie

Combining Differentiation and Challenge in Mathematics Instruction: A Case from Practice

Keywords: Individualised instruction, Productive struggle, Problem sets, Difficulty level, Teaching methods, Classroom environment

Despite the challenges of teaching mathematics to diverse learners and the importance of students engaging in cognitively demanding tasks, practice-based research on instruction that combines these priorities is relatively rare. In this paper, a primary mathematics laboratory is used as the setting for implementing such instruction. Records of practiceincluding video records of teaching, lesson plans, and student workare analysed to identify task characteristics and teacher actions that supported differentiated instruction and work on challenging tasks. Among the themes identified are the teacher actively guiding the students' learning, the establishment of classroom norms, and enabling all students to act as resources for one another's learning.

Introduction.

In many countries classrooms are increasingly diverse and teachers need to cater for students from a wide variety of intellectual, linguistic, cultural, social and economic backgrounds. However, relatively few supports are available to help teachers differentiate instruction so that all students, despite their differences, can meaningfully participate in mathematics lessons. Furthermore, using challenging tasks has the potential to improve students' learning (e.g. [START_REF] Stein | Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms[END_REF] by engaging students in productive struggle [START_REF] Warshauer | Productive struggle in middle school mathematics classrooms[END_REF]. One problem is that sometimes in classroom practice only high-achieving students are challenged or differentiated instruction is intended only for some students. Until relatively recently, few studies combined a focus on challenging students mathematically while simultaneously differentiating instruction (e.g. [START_REF] Brown | Using push and support cards for differentiation[END_REF][START_REF] Lynch | Productive struggle for all: Differentiated instruction[END_REF][START_REF] Sullivan | Classroom culture, challenging mathematical tasks, and student persistence in[END_REF]. This paper presents a case study of one teacher attempting to engage students in cognitively demanding work over the course of a lesson while simultaneously differentiating instruction.

The following research question is addressed: What steps taken by the teacher supported differentiation and maintained or modified the challenge of the task?

Theoretical Framework.

Two areas of research inform this study. The first is the mathematical task framework proposed by [START_REF] Stein | Mathematical tasks as a framework for reflection: From research to practice[END_REF] which classifies mathematical tasks into those with lower-level demands (consisting of memorisation tasks and procedures without connections) and higher-level demands (consisting of procedures with connections and doing mathematics). This framework was necessary to identify times when the teacher maintained, raised or lowered the cognitive demand of the task. Although specific tasks may be initially classified using these categories, the classification may change over the course of instruction, as the teacher sets up the task for students and as the task is implemented in the classroom. Tomlinson's work (2000) on differentiating instruction provides the second pillar of the framework because the teacher's goal is for all students to experience challenge in line with their readiness for it. Tomlinson identifies four elements which teachers can differentiate to meet the needs of all students: content, process, product, learning environment. Content of tasks can be differentiated by presenting information to students in multiple communication modes such as written or oral, or by providing supports to help them read the text. Differentiating the process allows students to complete a task in various ways depending on their preferences; this may involve providing materials, varying the time allocated, permitting individual or group effort, or breaking a task into smaller parts for some students. The product may be differentiated if students are allowed to present their work in multiple ways and to receive credit for diverse skills that are evident in the completed product. Finally, the learning environment may be differentiated by providing quiet space in the classroom for some students, being explicit about how help may be accessed if a student is stuck, and making available culturally familiar support materials to students as necessary.

The twin lenses provided by combining insights from [START_REF] Stein | Mathematical tasks as a framework for reflection: From research to practice[END_REF] and [START_REF] Tomlinson | Differentiation of instruction in the elementary grades: ERIC Digest[END_REF] will help to identify steps taken by the teacher to address both goals of instruction: productive struggle and differentiation.

Method.

Context.

The study is based on a mathematics laboratory that took place over four days (approximately eight hours) in July 2018. The classroom was convened specially to provide mathematics teaching for students, with a focus on differentiated instruction and challenging all students. Furthermore, the laboratory was a site for professional development with over 20 researchers, teachers and teacher educators observing the teaching and subsequently discussing it. Finally, the laboratory was a site for research with extensive records of practice being collected, including video recordings, student work, other classroom artifacts, and teacher planning notes.

Participants.

Twenty-four children -10 girls and 14 boyswere in the class which lasted for two hours per day over four days. The children had completed fifth class in 11 different schools. Parents and guardians applied for places in the school for their children for multiple reasons, such as their child finding mathematics difficult, their child liking mathematics, or because a friend was attending. In other words, although parents opted to send their child to this class, a wide range of levels of mathematics and motivation could reasonably be expected among those participating.

The teacher has substantial experience as a primary teacher (11 years) and a primary mathematics teacher educator (19 years). He works as a researcher on a project that is focused on mathematics instruction that is differentiated and challenging for students. For several years he has taught children mathematics in a laboratory format as a form of professional development for teachers but this was the first time that the focus was explicitly on differentiation and challenge.

Over the four days of the laboratory, the students worked on four mathematical tasks, each for two hours. One was based on volume, two on fractions, and one, the subject of this paper, on algebra (see Figure 1).

Data.

Four sources of data were used in this study. First, the task itself, which went through several iterations before being used. Second, the lesson plan that was used as the basis of the lesson. Third, a video recording of the lesson as it was enacted will be analysed. Finally, students' work on the task will be presented.

Data analysis.

Open coding in which data are scrutinised and concepts identified [START_REF] Corbin | Basics of qualitative research: Techniques and procedures for developing grounded theory[END_REF] was used for analysing the four data sources. This involved closely and repeatedly reading the written materials and closely watching and re-watching the video data. The second author identified initial categories and themes, using methods outlined by [START_REF] Corbin | Basics of qualitative research: Techniques and procedures for developing grounded theory[END_REF]. Although the coding was open, in that the data were decomposed and concepts were sought and named, the coding complemented a deductive approach to data analysis [START_REF] Gale | Using the framework method for the analysis of qualitative data in multi-disciplinary health research[END_REF] which was informed by the frameworks of [START_REF] Stein | Mathematical tasks as a framework for reflection: From research to practice[END_REF] and [START_REF] Tomlinson | Differentiation of instruction in the elementary grades: ERIC Digest[END_REF]. Findings.

In terms of the framework/task analysis guide by [START_REF] Stein | Mathematical tasks as a framework for reflection: From research to practice[END_REF], the chosen task is deemed to make "higher level demands" on learners and it has characteristics of both procedures with connections (e.g. use of a table for the first bullet point; writing a rule for the second; getting students to compare solutions for the third) and doing mathematics.

In analysing video recordings of the two hours of teaching, three themes were identified: Teacher actively guiding the students, teacher establishing classroom norms that facilitated differentiation and work on challenging tasks, teacher facilitating students to be resources for each other's learning. Each of these themes will now be elaborated on in more detail.

Teacher actively guiding the students.

Six ways in which the teacher actively guided the students were identified by analysing the tasks used, the lesson plans and the video. The first related to the choice of task and planning. Each task was chosen carefully so that all students had the opportunity to make some progress on it and some students could complete it all. For example, the basic algebra task (See Figure 1) was complemented by preliminary "enablers", for students who struggled to start or make progress on the task and more challenging "extensions", for students who successfully completed the basic task.

One enabler suggested organizing data in a table and another commenced a rule pattern which could be continued. An extender presented for comparison three different methods used by other "students" to find the rule to calculate the number of squares needed for any size chair.

During autonomous work, when two students were working on tasks, clear instructions were issued to students. For example, the teacher suggested that "to make things easier for yourselves, make sure you label the different chairs."

The teacher deliberately directed students' attention to particular ideas or concepts. For example, at different times he found ways to introduce the idea of a row, a constant, and a variable. These concepts were introduced in response to comments or confusions from the students and would be help them in working towards a general rule.

The teacher sought opportunities to highlight ideas he believed would support students' work. As students worked on the task, the teacher asked them to pause their work and he introduced the following idea to support their work:

Just one thing I want to show you. This is just a convention and it might be helpful for some of you. Some of you have already moved on to this step. In algebra, which is what you're doing now, this is called algebra, this kind of maths. One of the things you do in algebra sometimes is that you use letters as well as numbers so sometimes you might use x and y. Sometimes you might use a and b or sometimes you might use n." (5533 @23:32)

In Lilly's work (Figure 2), it appears from what she has written that she sees the pattern. However, although she is aware that the letters may be used, she is unable to integrate them into the general solution that she has identified. In contrast, Abby was able to express one general formula that could be used to calculate the number of tiles in a chair, where y is the size of the chair and y x 3 +5

gives the number of tiles (Figure 3). Note that Abby also uses the term "constant" in her written answer.

The teacher questioned students for clarification. For example, using Lilly's idea of a chair of size 0, he asked "Can anyone explain what Lily has just said or put it into your own words?" (5527@8:40). On another occasion he used revoicing [START_REF] O'connor | When is dialogue dialogic?[END_REF], when he attempted to clarify what a student was saying, "so you're saying the shape is always the same except for…." (5534@19:12).

The teacher attempted to increase the challenge for students. For example, he asked if anyone could come up with a rule that works for any size chair (video clip 5527, 5:03) and on another occasion he set the following challenge for students: "Now I want everyone to think carefully. How does David's rule compare with Tim's rule? (Clip 3527 @ 15:55).

Finally, the teacher's active guiding of students involved giving them time to think. After posing a question, he allowed students sufficient time to think of an answer before calling on someone to answer.

Teacher establishing classroom norms that facilitated differentiation and work on challenging tasks.

A second theme identified relates to how the teacher established and promoted classroom norms that facilitated differentiation and work on challenging tasks. One such norm that was encouraged was that confusion is okay on the path to learning/understanding. At one point the teacher said "Remember it's really important to say that you're confused. Have you followed anything at all of what Tim has said?" (5527@11:05). This followed a contribution from Tim that would lead towards a solution to the task.

Other norms evident in the video include the use of talking about mathematics. Students do not just write their solutions but they are encouraged to share their ideas with their classmates and to have them questioned and critiqued by their peers.

Students are encouraged to take responsibility for their learning. The nature of learning is discussed. For example, students are told that like working out in the gym, learning mathematics requires effort. That doesn't mean that "maths is difficult" or you are "no good at maths" but that over time, the effort will lead to learning.

Manipulative materials are made available to students (e.g. square tiles) and students are encouraged to use what they need if using the materials will help them grasp the concept. Students are given autonomy in deciding to use or not use the materials.

Finally, students are encouraged to represent their solutions in visual form. In particular, they are encouraged to explain ideas for their parents using representations. This differs from a more common approach in schools where students simply present their solutions in their own notebooks in numerical form.

Teacher facilitating students to be resources for each other's learning.

The third theme identified from the video data was the teacher's encouragement of students to be resources for one another's learning. At the outset of the summer school the teacher explicitly stated to the students that "you can learn a lot from your classmates." The idea of students being resources for each other's mathematics learning was encouraged throughout the summer school.

Figure 2: Lilly's work on the algebra problem

One way the teacher did this was to ensure that students or groups of students were given opportunities to share their solutions. Attempts were made to sequence the sharing in a way that progressed from more naïve to more sophisticated solution strategies.

Students' sharing of solutions was complemented by several examples of the teacher asking students to repeat, revoice, or explain what was said. This had the potential to amplify what students said (by hearing them a second time), to clarify what was said (through explanations) and it encouraged students to listen to one another because they realise that they may be asked to respond to another student's input.

Students were required not only to listen to contributions from their classmates; the teacher frequently asked students to analyse and compare different solutions and ideas. This affirms students' solutions but it further has the potential to help them see where some solutions might be more efficient than others.

Finally, students are frequently asked to clarify their ideas or those of their peers by using words and representations. For example, on one occasion when students were working on the task, the teacher asked a student to "Come up and show us how you laid it out in your notebook" (5527 @9:20). We return now to the research question that we posed at the start of this paper about how the teacher supported differentiation and maintained or modified the challenge of the task. Modification of the task was planned in advance through the use of enablers and extenders. In the course of the lesson three themes were identified to capture ways in which the teacher implemented differentiation and cognitive demand. He actively guided the students to take responsibility for working on the task. He established and maintained classroom norms that supported these aims and he encouraged students to act as resources for each other's learning.

What these findings suggest is that through intentional and deliberate actions, a teacher can combine both differentiation and challenge in mathematics instruction. The need for differentiated instruction may be imposed on a teacher by the different rates at which students complete tasks. However, in order to engage students in challenging tasks, teachers must set that as a specific instructional goal and must have the required knowledge to implement it [START_REF] Henningsen | Mathematical tasks and student cognition: Classroombased factors that support and inhibit high-level mathematical thinking and reasoning[END_REF]. Although the importance of teacher guidance and establishing appropriate norms have been identified as factors in supporting the use of challenging tasks independently, the additional factor of students acting as resources for one another was found to apply when challenge was combined with differentiated instruction.

  Alex uses identical tiles to make different sized chair designs for a school art project. The pictures on the sheet show the first three designs created, size 2, size 3 and size 4. Alex wanted a rule that would help work out the number of tiles needed for a chair of any size. Q 1 (a) If Alex wanted to create a size 5 chair, what would it look like? Can you draw it or use other materials to represent it? How many tiles would be used? (b) Work out the number of tiles needed for the size 6 and size 7 chairs. Explain how you did this. (c) Draw or make the size 1 chair. How many tiles did you need? Q 2 (d) Do you notice any pattern between the chair size and the number of tiles needed each time? Discuss this pattern with your partner(s). Q 3 (e) Alex wanted to create a size 20 chair. Talk with your partner(s) about a rule that would help Alex work out the number of tiles needed for this chair. (f) Would this rule work for the previous chair sizes? (g) If yes, write out this rule in words. (h) Discuss if it would work for a chair of any size. Q 4 (i) Could you re-write this rule using symbols/letters? Q 5 (j) Use the rule to calculate the number of tiles needed for a "size 50" chair? Figure 1: Algebra task (without enablers or extensions)

Figure

  Figure 3: Abby's work on the algebra problem


Anneli Dyrvold
email: anneli.dyrvold@edu.uu.se

Ida Bergvall
email: ida.bergvall@edu.uu.se

Ida Bergvall Designing

Designing tasks with self-explanation prompts

Keywords: Scaffolding, ZPD, reading, multimodal, review

come

Introduction

In mathematics classrooms, individual and group work on mathematical tasks is a frequent activity. Besides the mathematical complexity of the task, different design aspects also have bearing on students' learning. In this paper, we focus on design aspects concerning self-explanation prompts (hereafter SEPs), and how they can be used in tasks or instructions to scaffold students' learning.

The use of SEPs has been investigated in previous studies, often with pre-and post-tests, and some design aspects regarding SEPs have proven to be more effective than others when it comes to students learning (e.g., [START_REF] Berthold | Assisting self-explanation prompts are more effective than open prompts when learning with multiple representations[END_REF][START_REF] Lin | Effects of visual cues and selfexplanation prompts: empirical evidence in a multimedia environment[END_REF][START_REF] Rau | Successful learning with multiple graphical representations and self-explanation prompts[END_REF]. In this study, we utilize a qualitative perspective in order to highlight aspects concerning how tasks or instructions with self-explanation prompts can be successfully designed and used to scaffold students' learning in mathematics in relation to the zone of proximal development (ZPD). The study is based on an ongoing literature review of previous empirical studies on SEPs and their effectiveness. We also take a broad stance and include studies on SEPs in different subjects. A bottom-up perspective is used to search for prominent characteristics in the sample of studies on SEPs. More precisely, the following research question is answered in the current paper: Which important aspects are there to consider in task design in order to scaffold students' learning in mathematics aided by self-explanation prompts?

Studies on task design refer to either domain specific or domain transcendent aspects (Thanheiser, 2017). Domain specific aspects concern for example how tasks can be designed to support conceptual understanding. Domain transcendent aspects on the other hand takes a more interdisciplinary stance on, for example, reasoning, life skills, or reading of multimodal texts. In this paper, we focus on domain specific aspects with a focus on pupils' development of conceptual understanding, as well as on domain transcendent aspects, mainly regarding how SEPs can be used to support students' ability to read multimodal mathematical texts. In this study the emphasis is laid on the resources natural language, images, and mathematical notation.

Scaffolding and self-explanation prompts

Scaffolding is theoretically founded in a socio-cultural tradition. It is a metaphor which illustrates how people are assisted to reach learning goals that would not easily have been reached without scaffolding. In accordance with scaffolding theory, SEPs are meant to support learning in the zone of proximal development (ZPD) and therefore scaffolding needs to be adapted to the pupil and his or her needs and level of understanding. Another aspect of the scaffolding metaphor is that when the building is finished, that is, when the student has developed the targeted knowledge, the scaffolding needs to be removed. Finally, the scaffolding includes a transfer of responsibility from the teacher to the student [START_REF] Bakker | Scaffolding and dialogic teaching in mathematics education: introduction and review[END_REF].

Traditionally, scaffolding has been referred to as the interaction between teacher and student, but the term was expanded to also include, for example, artefacts or instruction plans [START_REF] Bakker | Scaffolding and dialogic teaching in mathematics education: introduction and review[END_REF]. In the present paper, we study SEPs as a particular form of scaffolding of importance for task design.

In several previous studies, self-explanations are described as a successful means to increase students' knowledge and ability to solve problems (e.g., [START_REF] Chi | Eliciting self-explanations improves understanding[END_REF][START_REF] Rittle-Johnson | Promoting transfer: Effects of self-explanation and direct instruction[END_REF]. The self-explanations are explanations of a concept, a relation, or procedure given by the student to him or herself. SEPs has proved especially effective in subject areas such as mathematics and science, which often consist of general principles with few exceptions [START_REF] Rittle-Johnson | Eliciting explanations: Constraints on when selfexplanation aids learning[END_REF]. By explaining to oneself, it is possible to reconcile new information and to make inferences to prior knowledge [START_REF] Chi | Eliciting self-explanations improves understanding[END_REF][START_REF] Berthold | Assisting self-explanation prompts are more effective than open prompts when learning with multiple representations[END_REF]. What gives effect is the process to formulate an explanation, either to oneself or to others. Self-explanations however, are nothing that students usually do spontaneously and therefore prompts to self-explain have a large potential.

Method

The search for articles for the review has been limited to articles published after 2009. We both searched ERIC, with the search terms self, explain, and prompt and forward tracked selected key papers. Criteria for inclusion was a focus on prompts requesting students to self-explain while working on a typical written task or expository text. The collection of 41 relevant articles was analysed in an iterative process where categories were created and re-created based on what was revealed in the articles. Initially the articles were coded regarding choice of variables, method, and results, with a particular focus on how SEPs were defined and implemented. New categories both emerged and were dismissed based on what was eminent in the articles. The use of this iterative analysis resulted in a few final themes for which rich qualitative data could be derived from several of the analysed studies. Three of these themes are presented in the current paper.

Results

The answer to the question about which important aspects there are to consider in task design in order to scaffold student's learning in mathematics aided by SEPs is presented in three themes.

Theme 1: Using prompts for different purposes

Among studies that reveal positive effects of self-explanations, the prompts are used for three different purposes. Firstly, SEPs can scaffold the students in how to process the content and therefore enhance learning concerning domain specific aspects. The most prominent purpose is to encourage inferences. Inferences are provoked by SEPs including a why question (e.g., [START_REF] Roelle | The expertise reversal effect in prompting focused processing of instructional explanations[END_REF][START_REF] Nokes | Testing the instructional fit hypothesis: the case of self-explanation prompts[END_REF] or by explicitly requesting arguments [START_REF] Berthold | The double-edged effects of explanation prompts[END_REF]. The advantage with SEPs that provoke inferences has also been revealed in comparisons with other SEPs ( oelle ller, [START_REF] Roelle | Learning from Instructional Explanations: Effects of Prompts Based on the Active-Constructive-Interactive Framework[END_REF][START_REF] Neubrand | Tackling the difficulties in learning evolution: effects of adaptive self-explanation prompts[END_REF]. Inference prompts are superior for the acquisition of conceptual knowledge. It is evident that the active construction of the selfexplanation is intrinsic since other prompts given in combination with explanations of the inferences requested by the inference prompts are less beneficial [START_REF] Roelle | Learning from Instructional Explanations: Effects of Prompts Based on the Active-Constructive-Interactive Framework[END_REF].

Secondly, SEPs can be used in a domain transcendent manner to support reading, and therefore enhance learning. SEPs focusing on reading have shown positive effects on learning outcomes when the SEPs are designed to induce focused processing of the text and to avoid shallow reading. For example, SEPs providing reading guidance in terms of higher order questions that require the learner to actively generate inferences about unfamiliar content, have proven to be effective means to develop conceptual knowledge [START_REF] Roelle | The expertise reversal effect in prompting focused processing of instructional explanations[END_REF]. Gap-filling prompts that support reading by prompting the student to make inferences and by that add coherence to the text, also lead to greater learning, as shown by [START_REF] Nokes | Testing the instructional fit hypothesis: the case of self-explanation prompts[END_REF]. Particularly effective are step focused prompts, supporting students in the reading by drawing attention to each step in an example, with prompts to explain, elaborate and summarize (ibid.). Studies that evaluate the use of SEPs in multimodal text reveal different benefits in relation to the use of such prompts. When reading multimodal text, it is intrinsic to understand how the different representations relate and SEPs can successfully facilitate such an understanding. SEPs can scaffold the reading by prompting the reader to relate parts of, and explain relations within the text [START_REF] Rau | Successful learning with multiple graphical representations and self-explanation prompts[END_REF] as exemplified in Figure 1.

Figure 1: Open-ended prompts to make inferences and to relate between modalities For example, the student can be prompted to self-explain which perceptual features of graphic representations that depict corresponding concepts and complementary information [START_REF] Rau | Successful learning with multiple graphical representations and self-explanation prompts[END_REF]. Such prompts to relate and translate between representations lead to better learning results compared to general prompts to self-explain (e.g. 'explain your answer') (van der eij & de Jong 2011). SEPs are particularly useful when many representations must be related to deeply understand key concepts [START_REF] Rau | Successful learning with multiple graphical representations and self-explanation prompts[END_REF]. Figure 1 is an example of how prompts can be used to support reading of multimodal texts. The first prompt draws attention to how the bars in two different colours represent the two classes mentioned in the text. The second prompt foster attention to the meaning of the axes in the graphic representation and demand the reader to relate information given in natural language and visually. The knowledge of how to read and understand the various parts of the diagram is essential.

Lastly, SEPs can diminish the cognitive load and therefore contribute to the fulfilment of the first two purposes (support reading and processing of the content). If SEPs are designed to guide the attention to relevant principles in a text, extraneous cognitive load can be decreased, and as a result, learning can be improved [START_REF] Wang | Do Focused Self-Explanation Prompts Overcome Seductive Details? A Multimedia Study[END_REF]. There is however a potential opposite effect; SEPs can increase extraneous cognitive load if they do not suit the reader, for example if the prior knowledge is high and the prompted inferences therefore are perceived as redundant and distracting.

Theme 2: Adaptation of prompts to students' prior knowledge

The positive outcomes from the work with SEPs is dependent on the match between the student and the prompt. In particular, the students' prior knowledge is crucial. For example, prompts to make inferences are demanding. [START_REF] Neubrand | Tackling the difficulties in learning evolution: effects of adaptive self-explanation prompts[END_REF], who categorizes inference building prompts as high-knowledge prompts, do however reveal that even for medium knowledge learners the most positive impact with regard to quality of SEPs is created by using a combination of less demanding prompts (paraphrasing, recourse of previously given information, searching for relevant relations between parts of the text) and high demanding prompts (anticipative, inference-building and with recourse to prior knowledge).

In some studies, interventions with SEPs do not lead to better learning outcomes than for the control group. One possible explanation for the lack of effect of SEPs is either that the knowledge level with abstract concepts is too high for the students [START_REF] Hsu | Examining the effects of combining self-explanation principles with an educational game on learning science concepts[END_REF], or that the level is too low and that the prompted self-explanations hinder the students in their work [START_REF] Roelle | The expertise reversal effect in prompting focused processing of instructional explanations[END_REF].

In a socio-cultural perspective, this can be understood as if the concepts are not held within the students' ZPD. If the new information is at an overly high level, the students will not have the opportunity to make the connections between new information and previous knowledge, which are required for learning. The intended effect of SEPs can also be missing if the level of the concepts or procedures that students should develop is too low. In a study by [START_REF] Neubrand | Adaptive prompts for learning evolution with worked examples-highlighting the students between the "Novices" and the "Experts" in a classroom[END_REF] it appeared that worked examples (WE) with SEPs are effective for students with medium-level of prior knowledge. For students with good knowledge of the subject, WE with SEPs were negative for learning outcomes, no matter how they were designed. SEPs then presents a disadvantage for the students since the previous knowledge and the information given in the WE simply becomes redundant and perhaps boring or distracting. Figure 2 shows a task with different prompt options which can be used for students at different levels of knowledge. The first version of a prompt is intended for students with less prior knowledge. Students who are very unfamiliar with this type of task can also be given a worked example with a starter prompt. Version 1 gives more scaffolding and gives recourse to representations in the task and is therefore adapted to less prior knowledge. Version 2 offers less scaffolding, and encourages the student to use prior knowledge while making inferences and is therefore adapted to medium to high prior knowledge.

Theme 3: Designing tasks and self-explanation prompts that engage students A crucial factor in achieving the intended outcome from the use of SEPs is student engagement in the explanations. In several studies (e.g., [START_REF] Hsu | Examining the effects of combining self-explanation principles with an educational game on learning science concepts[END_REF] the students' answers to the given SEPs were not sufficiently well processed and the quality was insufficient. However, the results in these studies also showed that students who actually produced well-developed and high-quality responses also showed good results in the post-test. The reasons why students did not engage in the SEPs varied. Pre-formulated self-explanations in a game environment for primary school children is one example of SEPs that was not sufficiently engaging for students to give any effect on learning [START_REF] Hsu | Examining the effects of combining self-explanation principles with an educational game on learning science concepts[END_REF]. Another reason for students not to engage actively in the work with SEPs was that the students experienced the SEPs as demanding and simply skipped them [START_REF] Lin | Effects of visual cues and selfexplanation prompts: empirical evidence in a multimedia environment[END_REF].

Students' lack of engagement in SEPs can also be explained by the extensiveness of the SEPs. In a study by [START_REF] Kapli | The Effects of Segmented Multimedia Worked Examples and Self-Explanations on Acquisition of Conceptual Knowledge and Problem-Solving Performance in an Undergraduate Engineering Course[END_REF], no effect on conceptual knowledge or problem solving performance were found if SEPs were given together with supportive instructions. The scaffolding given by the SEPs was however not gradually faded out, rather the scaffolding was continuously accessible for the students. This may have had the effect that students did not take sufficient effort to understand the concepts; instead, they could use the support at any time. However, the study also showed a positive correlation between quality in the students' self-explanations and the acquisition of conceptual knowledge.

Figure 2 is sufficient as an example of design aspects in relation to student engagement, since there is a relation between prior knowledge and student engagement in a task. With prompt options a student can choose no prompt, or prompts adjusted to more or less prior knowledge and to their willingness to engage in deep learning in a particular task. In accordance with theory about scaffolding, students are supposed to learn to manage by themselves what the prompts scaffold and therefore one optional version of a prompt is not always sufficient. In Figure 1, the student needs to take an active role in constructing the knowledge requested by the SEPs, which might be engaging. If the prompts on the other hand were formulated as multiple choice alternatives the prompts are likely to be less engaging (see e.g., [START_REF] Hsu | Examining the effects of combining self-explanation principles with an educational game on learning science concepts[END_REF].

Discussion

The current paper contributes to previous research on SEPs with a qualitative perspective and a focus on design aspects. The design of tasks with SEPs is done in relation to a target group, but the usefulness of a particular prompt can vary between students in a class and therefore our results is applicable also in relation to the implementation of the prompted tasks. We argue the usefulness of SEPs is dependent on a match between the task, the prompt, and the student. This interrelation is relevant in relation to all three themes presented in the paper.

It is also apparent that the three presented themes are related. Theme 1 about the prompt's purpose is related to both the other themes since the purpose must be taken into account when prior knowledge and students' engagement are considered. Theme 2 and theme 3 are also interrelated since engagement is often dependent on prior knowledge. This means the adaptation to all three themes in design of SEPs could be highly beneficial in providing opportunities for learning, when they are combined and taken into account simultaneously. Figure 3 illustrates how the three themes overlap in a common intersection. According to the results of this study, tasks that can be described as belonging in this intersection, offer the best opportunities for learning. Secondly, the SEPs must be adapted to students' prior knowledge that is relevant in relation to the content of the mathematics text. Since prior knowledge vary in a group of students we argue for a flexibility in the use of SEPs, both by utilizing variants of SEPs (Figure 2) and by encouraging students to be flexible in the use of SEPs in such a way that they will only be used when needed. As the students develop their understanding, the SEPs need to be taken away since scaffolding shall be seen as a temporary support when new abilities and knowledge are developed [START_REF] Bakker | Scaffolding and dialogic teaching in mathematics education: introduction and review[END_REF]. Such an adaptation to the students is important also since redundant SEPs may increase the cognitive load or distract students and accordingly the teacher needs awareness of this.

Thirdly, SEPs are effective only if students engage thoroughly in the self-explanations. As mentioned previously the fit between students' knowledge and the SEPs is crucial if the SEPs shall lead to the intended learning [START_REF] Neubrand | Tackling the difficulties in learning evolution: effects of adaptive self-explanation prompts[END_REF] and too demanding or too trivial prompts can reduce students' commitment to the SEPs. SEPs to make inferences are demanding but also very efficient. An implication for teaching is thus that students need to be given the opportunity to practice more demanding self-explanation in classroom. Students may need to practice on how to formulate appropriate answers to these types of prompts, and to get feedback on their answers. In this way, students can understand what is expected and also understand that the answers are important, which can motivate students to work with their answers to make them high quality.

Lastly, when working with SEPs it is important to bear in mind that the scaffolding has to fade out as students' knowledge increase. This can be done in different ways. In teaching materials, scaffolding ought to be used in tasks initiating a new topic or concept and in such occasions the scaffolding fills a function, for example, in supporting reading of multimodal texts. As the teaching proceeds and the students improve their ability to read multimodal text, the scaffolding become superfluous.

It is also possible for the teacher to take an active role in the use of scaffolding, by carefully following the development of the students' knowledge in order to use SEPs only to students considered in Theme 1

Theme 3 Theme 2 need for this support. To conclude, we see a large potential in the use of SEPs but the design of prompts and the adjustments to the student group is a delicate task to achieve the intended learning.

  The pupils in two classes investigated how many siblings they had. The result of the survey is presented in the diagram below. many pupils in class A took part in the survey? b) How many pupils have three siblings? c) Determine the mean and the median of the number of siblings in class B.Before you do the task, complete the following starter prompts:1. One reason the two different colors of the pillars are helpful in reading the diagram is … 2. The labels to the horizontal and the vertical axis mean that the pillars shows …

a)Figure 2 :

 2 Figure 2: Task with two versions of self-explanation prompts

Figure 3 :

 3 Figure 3: Three interrelated themes, appropriate to consider togetherIn the everyday practice when using SEPs in mathematics there is several aspects to consider. Firstly, SEPs can scaffold different types of learning and the teacher, or task designer, must know what he or she intends to achieve with SEPs. Students may benefit from SEPs when new concepts are introduced since the SEPs support them in integrating knowledge to what they have learnt previously (e.g.,[START_REF] Nokes | Testing the instructional fit hypothesis: the case of self-explanation prompts[END_REF] and from SEPs that support reading text with different representations of a concept (e.g.,[START_REF] Rau | Successful learning with multiple graphical representations and self-explanation prompts[END_REF].