Adrien Deloro
email: adrien.deloro@imj-prg.fr

Joshua Wiscons
email: joshua.wiscons@csus.edu

The geometry of involutions in ranked groups with a ti-subgroup

Solomon saith, There is no new thing upon the earth. So that as Plato had an imagination, That all knowledge was but remembrance; so Solomon giveth his sentence, That all novelty is but oblivion. Whereby you may see that the river of Lethe runneth as well above ground as below.

Introduction

The present research belongs to the intersection of geometric algebra and model theory. Our work follows two major themes in the theory of groups of Lie type, i.e. the analysis of transformation groups of linear geometries:

• the study of the distribution of involutions (elements of order 2), strongly real elements (products of two elements), and their centralisers, as made systematic by Brauer and his school and exemplified by the classification of the finite simple groups;

• the identification of inner geometries, for instance incidence geometries which are already present "inside" the group (as opposed to representation theory where the geometry is some external, additional structure, or Chevalley theory where the Lie algebra and adjoint action are given).

Our context is model-theoretic, namely that of groups of finite Morley rank, generalising the familiar setting of algebraic groups over algebraically closed fields.

A theorem and conjecture

Two forms of one group and a conjecture. We investigate a family of configurations capturing two groups with much geometry in common: SO3(R) and PGL2(C). Consider SO3(R). The set I of its involutions is exactly that of half-turns of the space; each i ∈ I has an axis Li and a rotation plane Pi = L ⊥ i . Also note that distinct involutions i and j commute if and only if Li is contained in Pj (and dually). Thus, introducing the relation (i j) ⇐⇒ (ij = ji = 1), one finds that the incidence structure (I, I, ) is naturally isomorphic to the projective plane P 2 (R) = ({vector lines}, {vector planes}, ≤). As such SO3(R) as an abstract group defines P 2 (R) as an incidence geometry, a point which begs for several remarks.

1. The notion of definability is first-order definability, as made precise by model theory.

2. As first realised by Hilbert, the incidence structure P 2 (R) itself defines (a field isomorphic to) R; hence the group SO3(R) defines the field R, with all model-theoretic consequences, including that SO3(R) does not have finite Morley rank.

3. Treating involutions both as lines and dual planes highlights that there is an underlying polarity of the incidence structure.

4. Hilbert's coordinatisation theorem holds exactly of arguesian projective planes, viz. those satisfying the Desargues 10-point axiom. However, every projective plane which embeds into a projective 3-dimensional space is always arguesian, so if possible, identifying a 3-dimensional geometry is quite preferable to a 2-dimensional one. This is the point of view we shall adopt in the sequel. Incidentally, SO3(R) can indeed be endowed with the structure of a projective 3-dimensional space (inducing the above projective 2-dimensional structure on I), exactly as we define it in a more general setting below.

We now give the group-theoretic description of the phenomena. Let G = SO3(R) and C SO2(R) be the subgroup of rotations with given axis, say Re1. Letting i be the half-turn around Re1, we see that C = C • G (i) (the connected component of the centraliser). Every half-turn w with axis orthogonal to Re1 also commutes with i. Moreover, the action of w is a reflection of the plane Pi = Span(e2, e3), so conjugation by w inverts C. Hence CG(i) is not connected, and one finds that CG(i) = C w . Moreover, the set of conjugates of C forms a partition of G. Indeed, since every non-trivial rotation has a unique axis and G is transitive on P 2 (R), C intersects its distinct conjugates trivially, and the conjugates cover G. In symbols, G = g∈G/N G (C) C g , and in words, C is a ti-subgroup whose distinct conjugates cover G.

As a matter of fact and at the core of this paper, everything above almost holds in G = PGL2(C). Let C be the group of (classes of) diagonal matrices, which contains a unique involution i. Note that C is inverted by the anti-diagonal involution w, so as before, CG(i) = C w . Here again, distinct conjugates of C meet trivially, but now the resulting disjoint union only contains a generic subset of G, not all of it. That is, C is a ti-subgroup whose conjugates almost cover G. And returning to the geometry of involutions, in PGL2(C) the geometry is that of a generically defined projective plane: generic (but not all) pairs of points are connected by a unique line, and dually.

Of course, one should expect that SO3(R) and PGL2(C) have much in common: they are two forms of the simple algebraic group A1 (over reasonable fields). We suspect that the above geometry of involutions, which exists (perhaps only in a generically defined form) for groups possessing an almost self-normalizing ti-subgroup, characterises decent forms of the root system A1. The formal conjecture is a bit more modestly stated and only targets the "characteristic not 2" case.

Conjecture ("A1-Conjecture"). Let G be a connected ranked group with involutions but with no infinite elementary abelian subgroup. Suppose that G has a definable, connected subgroup C < G which is ti and almost self-normalising. Suppose that C has even index in N = NG(C). Then G PGL2(K).

Ranked groups.

It remains to explain our technical framework, and why SO3(R) disappeared. We will focus on groups of finite Morley rank, also known as ranked groups, which is a class extending groups of K-points of algebraic groups in algebraically closed fields and which was first discovered by model theorists. This gives us a well-studied context, but has the side effect of ruling out SO3(R) (as mentioned above, a point that hinges on the fact that infinite fields of finite Morley rank are algebraically closed).

More precisely groups of finite Morley rank are groups in which the class of first-order definable sets bears a dimension function, called the rank, which behaves like the Zariski dimension in algebraic geometry. Groups in this class have a lot in common with groups of K-points, in particular they possess:

• an efficient notion of connectedness and of a connected component;

• an efficient notion of "largeness", called genericity: subsets of maximal rank.

We wish to stress that the methodological framework forbids number-theoretic, topological, geometric (in the differential/functorial/infinitesimal sense of the term), and Lie-theoretic techniques. It has however been conjectured by Cherlin and Zilber independently that infinite simple groups of finite Morley rank ought to be groups of K-points of simple algebraic groups in algebraically closed fields, a statement which is both deep and wide open. Later it was realised that the analogy with the classification of the finite simple groups goes well beyond a mere similarity of statements to include many methods as well, suggesting that involutions, strongly real elements, and their centralisers are key notions in the domain. But quite interestingly, the research we describe here does not build on the bulk of this effort known as the "Borovik programme for the classification of simple groups of finite Morley rank". A rough summary of what we shall need from the theory is in § 1.3; the monograph [START_REF] Borovik | Groups of finite Morley rank[END_REF] can serve both as a reference and an introduction to the topic.

For the present work, could one have elected another technical environment? This is unclear to us. The conjecture above does not aim at identifying an abstract group (which implies identifying a specific field as well), but a root system. Since SO3(R) is the archetype of our configurations, one may wish to work in a context encompassing both algebraically closed and real closed fields, or more model-theoretically put, both ranked and o-minimal worlds. The latter also has useful notions of connectedness and largeness; moreover, the o-minimal analogue of the Cherlin-Zilber conjecture is already known. This suggests that one could have a future look at a common methodological denominator of both settings, for instance dimensional universes as developed in [START_REF] Deloro | Linearisation in model theory[END_REF]. But for the moment we shall content ourselves with finite Morley rank, at the cost of no longer admitting SO3(R) in our scope.

A theorem. In this article, we prove Theorem A below, together with various corollaries given in § 1.2. Theorem A is perhaps a bit subtle. We will add context, but the list of corollaries may be the best way to gauge its strength.

The reader unfamiliar with ranked groups is encouraged to think of G as a group of Kpoints over some algebraically closed field, which comes equipped with a dimension function (here called rank) but no a priori field nor topology. However, the "characteristic" is still quite relevant, which we approach as follows. A group G is U ⊥ 2 (this stands for: no 2-unipotent subgroups) if it contains no infinite elementary abelian 2-group; this may be safely read as working in characteristic not 2. The other case has been successfully dealt with in [START_REF] Altınel | Simple groups of finite Morley rank[END_REF].

We need two further definitions specific to the configuration we are studying. A subgroup C < G is ti (this stands for: trivial intersections) if: (∀g) (C = C g → C ∩ C g = 1). Also, the subgroup C is almost self-normalising if it has finite index in its normalizer, which, importantly, implies that the union of conjugates of C has full rank in G.

Theorem A ("The Geometric Theorem"). Let G be a connected, U ⊥ 2 , ranked group with involutions. Suppose that G has a definable, connected subgroup C < G which is ti and almost self-normalising.

Then g∈G C g does not contain all strongly real elements of G.

Theorem A is a revisitation of the classical analysis by Nesin of specific SO3(R)-ish configurations [START_REF] Nesin | Nonsolvable groups of Morley rank 3[END_REF] in a ranked context. It is however new and quite more general; we believe it clarifies the picture greatly as it focuses on involutions, not on subgroups. Its name refers to the fact that such a configuration naturally defines a G-invariant point-line-plane incidence structure that is, in fact, a three-dimensional projective space, just like in SO3(R).

Analogues of Theorem A could be, and may well have been, considered in various familiar settings. For example, one may wonder if the theorem holds in the Lie context with the last sentence replaced by "If g∈G C g contains all strongly real elements of G, then G SO3(R)", but we are unaware of such a result. In the finite case, one could ask the following-notice the analogy with the classical Brauer-Fowler "|G| 1/3 " lemma, the proof of which deals with the worst case scenario where centralisers of involutions are disjoint.

Question. Let G be a finite group (simple of Lie type if necessary) with a ti-subgroup C. Suppose that g∈G/N G (C) C g has cardinal ≥ (1 -ε)|G|. Get an upper bound for the number of involutions/lower bound for the order of the centraliser of an involution.

However, the present article may be seen as praise of generic methods for infinite configurations, with loss of effectiveness as a counterpart; and we do not (nor intend to) address this question here.

Final remarks.

Theorem A is a promising first step towards the A1-conjecture, resolution of which (or more precisely a technical variant of it) would eliminate one of three critical, minimal cases of the Cherlin-Zilber conjecture left open by [START_REF] Deloro | Involutive automorphisms of N • • -groups of finite Morley rank[END_REF], namely "CiBo2". Also, it should not be missed that the reconstruction of projective geometries from decent forms of PGL2(K) is also the topic of [START_REF] Weisfeiler | On abstract monomorphisms of k-forms of PGL(2)[END_REF]; however, in our case, completing a partial (more specifically, generic) projective geometry into a genuine one is not expected to proceed as in Weisfeiler's work but rather using model-theoretic tools.

Finally, however tempting and faithful to the spirit of the Cherlin-Zilber Conjecture, it would be immodest to drop the assumption on N in the A1-conjecture: in the special "strongly embedded" case (see Proposition 1), we have no contradiction in sight for now. The worst possibility is of course when N = C: then one has to tackle "generically Frobenius" ranked groups, of which not enough is known, and of which CiBo1 [START_REF] Deloro | Involutive automorphisms of N • • -groups of finite Morley rank[END_REF] could well be an extraordinarily pathological case.

Corollaries

The proof of Theorem A is in § 2. Here we list some of its consequences, which range from classical to new. Corollaries A.1 and A.2 will be quickly derived in § 2.3; Theorem B and its Corollaries B.1 and B.2 will be proved in § 3. Difficulties are better appreciated in light of the fact that a ranked analogue of the Feit-Thompson "odd order" theorem is nowhere in sight, and might well not exist.

Corollary A.1. Let G be a connected, U ⊥ 2 ,

ranked group. Suppose that G has a definable, connected subgroup C < G whose conjugates partition G. Then G has no involutions.

Corollary A.1 is entirely new. It was our starting point; then we realised that it was enough to capture all generic and strongly real elements to force genuine covering. Its setting is not the same as Jaligot's "Full Frobenius groups" [START_REF] Jaligot | Full Frobenius Groups of Finite Morley Rank and the Feit-Thompson Theorem[END_REF], where malnormality (viz. (∀g) (g / ∈ C → C ∩ C g = 1)) was assumed, and absence of involutions proved after Delahan and Nesin. It is a trivial matter to show that if in Corollary A.1 one supposes in addition that C is soluble or contains no unipotent torsion, then it is self-normalising, hence malnormal. Such configurations have been revived by Wagner [START_REF] Wagner | Bad groups[END_REF] as a generalisation of the ill-named "bad groups", of which more will be said below.

Modulo their non-trivial but standard local analysis, one immediately retrieves Corollary A.2, a celebrated classic. (Announced in rank 3 by Cherlin [Che79, §5.2, Theorem 1] with a flawed proof, the result has an interesting story. In an unpublished preprint [START_REF] Borovik | Involyutsii v gruppakh s razmernost[END_REF] Borovik foresaw the importance of geometrisation; the geometric line was simultaneously followed by Borovik and Poizat in a Soviet-published, much delayed article [BP91, Theorem 5.4], and by Poizat in his self-published, fast-track, book [Poi87, § 3.8]; Nesin's approach in rank 3 [START_REF] Nesin | Nonsolvable groups of Morley rank 3[END_REF] was different and independent, and to be later generalised by Corredor.)

Corollary A.2 (in rank 3, Nesin [Nes89, Main Theorem (2)]; in general, Borovik-Poizat [BP91, Theorem 5.4] or Corredor [Cor89, Theorem 1(5)]).
Let G be an infinite simple ranked group whose proper, definable, connected subgroups are all nilpotent. Then G has no involutions.

Parenthetically let us observe that if, in the previous corollary, only the definable, connected, soluble subgroups are supposed to be nilpotent, then this is open. Recall that the Borel subgroups are the definable, connected, soluble subgroups that are maximal as such.

Conjecture. Let G be a simple ranked group whose Borel subgroups are all nilpotent. Then G has no involutions.

Such a configuration is sometimes referred to as a "bad group"; the intriguing terminology has varied over the years, so one could consider refraining from using it. Also notice that what really matters in these tight configurations is the impossibility to apply the Schur-Zilber method (for defining fields) inside Borel subgroups. One might therefore include such hypothetical objects in the wider class of "asomic groups", i.e. groups not interpreting an infinite field-always bearing in mind that other methods for constructing a field exist.

What prevents us at present from proving the last conjecture is essentially that, as opposed to the algebraic world, Borel subgroups need no longer be conjugate in definable subgroups of G. This contrasts with the following result derived from Theorem A with only marginal extra work and in which hereditarily conjugate means that every definable connected subgroup of G enjoys Borel conjugacy. (We did our best to avoid the assumption, and could not.) Theorem B (generalises Borovik-Burdges [BB08, Theorem 3.8]). Let G be a connected, U ⊥ 2 , ranked group in which all Borel subgroups are nilpotent and hereditarily conjugate. Then the (possibly trivial) 2-torsion of G is central.

From Theorem B, we easily obtain Borovik and Burdges' "linear theorem", based on the preliminary analysis by Poizat [START_REF] Poizat | Quelques modestes remarques à propos d'une conséquence inattendue d'un résultat surprenant de Monsieur Frank Olaf Wagner[END_REF] (and later Mustafin [START_REF] Mustafin | Structure des groupes linéaires définissables dans un corps de rang de Morley fini[END_REF]). Corollary B.1 was a significant (though unpublished) result; indeed, even in this "definably linear" context, which one would think much easier than the abstract ranked setting, the Cherlin-Zilber conjecture is open and there is no known analogue of the Feit-Thompson theorem there either. Our proof is fundamentally new.

Corollary B.1 (Borovik-Burdges, [BB08, Main Theorem]

). Let K be a ranked field of characteristic 0 and G ≤ GLn(K) be a simple, definable, non Zariski-closed subgroup. Then G has no involutions.

We also found something we were not looking for.

Corollary B.2.

Let G be a connected ranked group in which all Borel subgroups are good tori. Then the (possibly trivial) 2-torsion of G is central.

Remark.

The authors learnt shortly before first submitting that Altınel, Berkman, and Wagner had also obtained a very special case of Corollary B.2, when the Prüfer 2-rank is 1 [ABW18, Theorem 8]. Their methods are also geometric but seem to focus on the plane of involutions (and ultimately Bachmann's theorem [BN94, §8.3]) instead of the full 3-dimensional geometry that is present.

A self-guide to the present paper

We close this introduction by reminding our reader what they will need to deem familiar in order to follow the main lines of our arguments. More specific references will be provided at technical moments.

• Basic group-theoretic definitions such as a strongly real element [BN94, § 10.1] and a strongly embedded subgroup [BN94, § 10.5].

• Some rough notion of what a group of finite Morley rank is; the unfamiliar reader should systematically use the analogy with groups of K-points in algebraically closed fields, in which the Zariski dimension "ranks" the constructible class but there is no functorial nor topological information. Among our fundamentals are the additivity property of rank [BN94, § 4.1.2], enabling basic rank computations, and also the notion of connectedness [BN94, § 5.2] and why generic (i.e. full rank) subsets of a connected group must intersect. Something almost holds if it does up to taking connected components. At times we also need the notion of the definable envelope of a subset X ⊆ G, which we denote by X def , as opposed to [BN94, § 5.5] where it is called definable closure and denoted d(X).

• Very interestingly, classical and fundamental results such as the Chevalley-Zilber generation lemma ("indecomposability theorem" [BN94, Theorem 5.26]), the Schur-Zilber linearisation lemma ("field theorem" [BN94, Theorem 9.1]), and the analysis of groups of small Morley rank play no role at all. We would be glad to take it as an indication that our result goes beyond the mere theory of groups of finite Morley rank. But on the other hand we shall invoke Macintyre's theorem [BN94, Theorem 8.1] that an infinite field with a rank function is algebraically closed: something not true in other model-theoretic settings.

What really matters is torsion; torsion-lifting [BN94, Exercise 11 on p. 93], basic divisibility properties [BN94, Exercises 9 and 13 on p. 78], but also the following.

• Recall that a U ⊥ 2 group is one not containing an infinite elementary abelian 2-group. In the ranked case, Sylow 2-subgroups (which are as always conjugate [BN94, Theorem 10.11]) then become toral-by-finite [BN94, Corollary 6.22], i.e. finite extensions of (Z/2 ∞ Z) d , where Z/2 ∞ Z is the Prüfer quasi-cyclic 2-group. In a connected, U ⊥ 2 , ranked group with involutions, the integer d known as the Prüfer 2-rank is non-zero [START_REF] Borovik | Involutions in groups of finite Morley rank of degenerate type[END_REF].

• Torality principles from [START_REF] Burdges | Semisimple torsion in groups of finite Morley rank[END_REF] are key features: in particular, in a connected, U ⊥ 2 ranked group, every 2-element lies in a 2-torus.

• It is essential to know what a good or decent torus [START_REF] Cherlin | Good tori in groups of finite Morley rank[END_REF] is and, in particular, the generic behaviour of the Cartan subgroups CG(T ) and their conjugacy, as well as their connectedness [AB08, Theorem 1].

If such tools, and a few more to be cited during the technical arguments, were available in the o-minimal context, Proposition 1 might carry over to there. The crux of the argument is however not from model theory and relies on two deep theorems.

• A projective plane can be arguesian (i.e. satisfy the Desargues property) or not; a projective plane is arguesian iff it can be coordinatised by a skew-field (Hilbert); a projective 3-space is always arguesian. See [START_REF] Hartshorne | Foundations of projective geometry[END_REF] (more accurate references in due time).

• Borel's fixed point theorem [Hum75, Theorem 21.2]: provided K is an algebraically closed field, the group of K-points of a soluble, connected algebraic group acting on a complete algebraic variety must have a fixed point.

We note that the remarkable work by Bachmann and his school on the group-theoretic axiomatisation of Euclidean geometry (viz. incidence+metric) plays no part in our approach. We insist that their work, however beautiful, belongs to metric geometry: methodologically speaking, it can therefore be no part of a ranked study, where fields are algebraically closed.

Theorem A

Theorem A. Let G be a connected, U ⊥ 2 , ranked group with involutions. Suppose that G has a definable, connected subgroup C < G which is ti and almost self-normalising.

Then g∈G C g does not contain all strongly real elements.

The present section is devoted to proving Theorem A, with a word on Corollaries A.1 and A.2 in § 2.3. A brief outline of the proof itself is as follows, in the same notation as the statement.

• The main alternative (Proposition 1 of § 2.1) is highly restrictive. If C is ti and selfnormalising, then either NG(C) is strongly embedded, or G shares many features of groups of type A1 over C or R; which covers the structure of the Sylow 2-subgroup, the generic distribution of involutions, and even the generic nature of G.

• The assumption that G C g contains all strongly real elements actually rules out strong embedding, so Proposition 1 has much to say. But as opposed to PGL2(K), the configuration is now unviable: this is proved in § 2.2, Proposition 2. To find a contradiction we introduce a projective geometry; this is remarkably accelerated by going 3-dimensional with a built-in polarity, a strategy which must be compared with Nesin's work.

The main alternative

Notation. Throughout this subsection and the next, G will be a connected, U ⊥ 2 , ranked group with involutions, and C < G a definable, connected, ti and almost self-normalising subgroup.

We let N = NG(C) and I ⊆ G be the set of involutions; I • I is then the set of strongly real elements of G.

Proposition 1 (A French Nocturne). Let G be a connected, U ⊥ 2 , ranked group with involutions. Suppose that G has a definable, connected subgroup C < G which is ti and almost self-normalising.

Then the following are equivalent:

(i) N = NG(C) is not strongly embedded;
(ii) N/C has even order;

(iii) the Sylow 2-subgroup is as in PGL2(C) and for i ∈ I (the set of involutions), (vi) the generic element of G is strongly real (alt.:

C • G (i) is a conjugate of C; (iv) N = C Z/2Z (inversion
I • I is generic in G);
(vii) every strongly real element generic as such is in G C g (alt.: rk

I • I \ G C g < rk (I • I));
(viii) for i ∈ I, CG(i) is not connected;

(ix) if i, j is a generic pair of involutions, there is k ∈ I commuting with both.

Remarks.

• The reader may, like us, muse upon the fact that the first four conditions and (viii) provide structural descriptions; whereas, the other four yield generic information. Having such an equivalence is strong praise for generic methods in the study of groups of finite Morley rank.

• All are satisfied in PGL2(K) (where C K × is the algebraic torus, say the diagonal subgroup). Algebraically, SO3(R) has the properties (where C SO2(R) is the stabiliser of any non-zero vector in the natural representation), but its dimension function does not agree with the Morley rank. Return to the introduction for more details. The A1-conjecture is precisely that Proposition 1 characterises PGL2(K) among groups of finite Morley rank.

• In PGL2(K) and SO3(R) it also happens that I • I = G, i.e. every element is strongly real (in PGL2(K) every element is either unipotent or semi-simple, and strongly real in either case, while in SO3(R) every element is a rotation inverted by some half-turn of the space, hence strongly real). This property does not seem to follow from Proposition 1, although it would be a consequence of the A1-conjecture. And the latter does not look noticeably easier under the extra assumption that I • I = G.

• Knowing the mere structure of the Sylow 2-subgroup in (iii) does not seem to be enough. But [START_REF] Deloro | Involutive automorphisms of N • • -groups of finite Morley rank[END_REF] provides a thorough analysis of C • G (i) in N • • -groups, and we now say a (very optional) word about the CiBoes. In CiBo1, the (to our current knowledge, possibly ti) centraliser CG(i) = C • G (i) is strongly embedded. CiBo3 is irrelevant here: neither the centraliser of an involution nor that of a maximal 2-torus is ti. Under the assumption that its connected centralisers C • G (i) are genuinely ti, CiBo2 is a model of Proposition 1, making the A1-conjecture non-trivial.

• Proposition 1 easily implies that I • C is generic in G. The converse may fail, as shown by the possibly ti configuration of type CiBo1. To our surprise, genericity of I • C does not seem to be a relevant property.

• Non-connectedness of CG(i) implies C • G (i) = C, but here again the converse may fail. And there are no claims whatsoever on whether N = C or not in the strongly embedded case.

Proof. By an obvious rank computation, Γ = g∈G C g is generic in G. It follows that C contains the connected centraliser of each of its non-trivial elements:

if 1 = x ∈ C, then CG(x) ≤ NG(C) since C is ti, so C • G (x) ≤ N • G (C) = C by almost self-normalisation. Now, C also contains a maximal decent torus of G: letting T ≤ G be one such, we know that g∈G CG(T ) g is generic [Che05, Lemma 4]. Since so is Γ, up to conjugacy there is 1 = x ∈ C ∩ CG(T ), whence T ≤ C • G (x) ≤ C. Hence by torality [BC09, Theorem 3], every 2-element is in a conjugate of C.
Since C is ti, an element x = 1 belongs to at most one conjugate of C, which we then denote by Cx. As we just saw Cx is defined (at least) for x a generic element of G or a 2-element. We stress that Cx does not stand for C • G (x), but for the unique conjugate of C containing x; in the strongly embedded case equality could fail and all we know is C • G (x) ≤ Cx. There will be two main blocks: (i)-(vii) and (viii)-(ix). Actually we treat (viii) somehow separately, since the current status of generic use of the Borovik-Cartan polar decomposition in odd type is not fully clarified. This is why we shall prove: [(i)-(vii)]⇔(ix)⇔(viii). Of course (iv)⇒(viii), but we want to stress that equivalence of (i)-(vii) with (ix) does not rely on unpublished material. The proof begins. (iv)⇒(v). Clear.

(v)⇒(vi).

As we know, Γ = G C g is generic in G.

(vi)⇒(vii).

A strongly real element generic as such is then just a generic element of G; and as we know, Γ is generic in G.

(vii)⇒(i). Suppose (vii) and yet that N is strongly embedded. Let S = I • I be the set of strongly real elements and Σ = S ∩ Γ. The assumption is that rk(S \ Σ) < rk S, which implies rk Σ = rk S (the degree of S is a priori unknown; of course once S is proved generic, this will be settled at once). By strong embedding, all involutions in N are actually in C: this is because N conjugates its involutions [BN94, Theorem 10.19] and C = N • contains one, say i, with connected centraliser

D = C • G (i) ≤ C. Then the set of involutions of C is IC = i N ,
µ : IC × (I \ N ) → S.
We contend that it has trivial fibres, and that the image avoids Σ. The former uses essentially the same argument as the latter: if ir ∈ C g in obvious notation, then i, r ∈ N g which forces C = Ci = C g = Cr, a contradiction. Therefore:

rk IC + rk(I \ N ) = rk µ (IC × (I \ N )) ≤ rk(S \ Σ) < rk S.
(Notice that we used rk(S \ Σ) < rk S, not only rk Σ = rk S.) Thanks to rk(I \ N ) = rk I and G-conjugacy of involutions ([BN94, Theorem 10.19] again) we derive the following contradiction:

rk S > rk C -rk D + rk G -rk D ≥ rk Σ.
This shows that (i)-(vii) are equivalent.

(viii)⇒(ix).

Whether N is strongly embedded or not, we know by [BN94, Theorem 10.19], resp. the implication (i)⇒(iii) (and torality principles), that all involutions are conjugate, say I = i G . Since fibres of the map g → i g are translates of CG(i), there is a correspondence between the generic element of G and the generic involution (genericity over i). Moreover I 2 has degree 1. Now if for the generic pair of involutions (r, s) the definable envelope rs def (viz. the smallest definable subgroup containing rs) is 2-divisible, then for g ∈ G generic over i, so is ii g def . Thus, the generic version of the Borovik-Cartan polar decomposition applies, showing that CG(i) is connected: a contradiction. (The decomposition we refer to can be derived using the tools of [BBC07, § 5], but the anxious reader should note that (viii) is not used further in the present article.) (ix)⇒(viii). Suppose that CG(i) is connected. Let (i, j) be a generic pair of involutions. By (ix) there is k ∈ I commuting with both, so i ∈ CG

(k) = C • G (k) ≤ C k , forcing Ci = C k = Cj likewise,
violating genericity. So (viii) and (ix) are equivalent. We finally establish the connection with (i)-(vii).

(ix)⇒(i). Let (i, j) be a generic pair of involutions. By assumption there is k commuting with both; say

C k = C g i . Now if N is strongly embedded, then i ∈ Ci ∩ N (C k ) ≤ N (Ci)∩N (Ci) g implies that g ∈ N (Ci), whence Ci = C k = Cj
likewise, against genericity of (i, j).

(i)⇒(ix). We make free use of all properties (i)-(vii). Let i be the involution of C. Consider the restriction of the multiplication map:

µ : (N \ C) × (I \ N ) → G.
The fibres are trivial since if (k, r) = ( , s), in obvious notation, have the same image,

then 1 = k ∈ C is inverted by r: forcing r ∈ N , a contradiction. Since rk(N \ C) = rk C and rk(I \ N ) = rk G -rk C, we find that µ is generically onto G. Moreover a generic element (k, r) ∈ (N \ C) × (I \ N
) is mapped to one of G, and in particular there is g ∈ G such that kr ∈ C g . Then k, r centralise i g : the pair (k, r) has the required property, which carries through conjugation to any generic pair.

The inconsistent configuration

We now venture into the inconsistent: as opposed to Proposition 1 which covered some existing groups, the next deals with pathological configurations.

Proposition 2. Let G be a connected, U ⊥ 2 , ranked group with involutions. Suppose that G has a definable, connected subgroup C < G which is ti and almost self-normalising. Then the following are equivalent: (x) every strongly real element is in G C g ; (xi) if i, j are any involutions, there is k ∈ I commuting with both;

(xii) G = G C g ; (xiii) G does not exist.

Remarks.

• Here again, SO3(R) has all properties.

• Using Proposition 1 it is a straightforward consequence of (xii) that G = I • I; the latter is however not inconsistent as it holds in PGL2(K).

Proof. Each condition clearly implies at least one in Proposition 1, which we use freely.

Claim 1. We have equivalence of (x), (xi), and (xii).

Proof of Claim. (x)⇒(xi).

Let i = j be involutions. Then we may assume that ij ∈ C: so i and j normalise C and centralise its unique involution.

(xi)⇒(xii).

We cheat a bit as we first prove (x). Let x = ij be any strongly real element. By assumption, there is an involution k commuting with i and j, so x ∈ CG(k). As we know from Proposition 1, x is in C • G (k) or an involution, so in any case, x ∈ G C g . We now prove that I • I • I = I • I. Let i, j, k be any three involutions. By assumption, there is an involution r commuting with both i and j and an involution s commuting with both r and k. We may assume that r = i, j as otherwise ij is an involution or 1, which implies that ij • k ∈ I • I ∪ I ⊂ I • I (by Proposition 1). In particular, ij ∈ C • G (r). Now, if s = r, then k ∈ C(r), and there are two options: k = r or k inverts C • G (r). In the former case, jk is an involution or 1, forcing i • jk ∈ I • I. In the latter case, ijk ∈ I ∪ {1} since ij ∈ C • G (r). So we are left to consider when s = r. Of course, ijk = ijs • sk. On one hand, as s inverts ij ∈ C • G (r), ijs is an involution or 1; on the other hand, the same is true of sk. Hence I = I • I is a definable subgroup containing the generic element by Proposition 1: so

I • I = G ⊆ G C g . ♦
From now on we suppose (x)-(xii). First we strengthen (xi) as follows. Claim 2. Let i = j be two distinct involutions. Then there is a unique involution which commutes with and is distinct from both. Proof of Claim. If i and j do not commute then an involution commuting with both (this exists by (xi)) must be distinct from both. If i and j do commute then ij is again distinct from both. We must now prove uniqueness. So let i, j, k, be four pairwise distinct involutions such that k and commute with i and j; notice how perfectly symmetric the configuration is; we are after a contradiction. The relevant portion of the commuting-involution graph is as follows-the involution r, which is introduced below, may in fact be equal to one of the others.

i j k r
Let r be the unique involution in C • G (k ) = Cr. We see that i, j, k, all normalise Cr, so as at most one of them can equal r, we may suppose that i, j, k / ∈ Cr. Then by the structure of the Sylow 2-subgroup, the fact that {1, i, k, r} forms a four-group forces i = kr; likewise j = kr = i, a contradiction. ♦

And now let us introduce an incidence geometry. Notation 3. Let Γ be the incidence structure having:

• for points the elements of G,

• for lines the translates of the various conjugates of C,

• for planes the translates of I, and set-theoretic incidence relation (which is transitive).

Notice that G acts on the geometry Γ both on the left and on the right-as opposed to if we had focused on the mere plane of involutions I, where only one copy of G acts (by conjugation). Claim 4. The action of G = G × G on Γ is flag-transitive.

Proof of Claim. Indeed, G is clearly transitive on planes, and as the stabiliser of the plane I is the diagonal group G ∆ G (acting on I as G does by conjugation), this stabiliser is transitive on the lines of I, which are precisely those of the form (N \ C) g . Consequently, the joint stabiliser of I and (N \ C) is N in its conjugation action, which is easily seen to be transitive on the points of (N \ C) by 2-divisibility of C. ♦

We also introduce a polarity, which will have the effect of considerably accelerating the proof. Notation 5. Let Ň = N \ C and define an operation ⊥ on Γ as follows:

• if g is a point, let g ⊥ = gI; • if L = gC is a line, let (gC) ⊥ = g Ň ;
• if gI is a plane, let (gI) ⊥ = g. Claim 6. ⊥ is a well-defined polarity of Γ, compatible with the action of G.

Proof of Claim.

Clearly ⊥ is well-defined for a point. For a line, it amounts to having C Ň = Ň , an obvious claim. Finally, for a plane: if gI = hI then x = h -1 g is strongly real, inverted by all involutions, so every involution normalises Cx, which we take to be G if x = 1. In fact, as we now see that I • I ⊆ NG(Cx), we get Cx = G and x = 1.

Of course ⊥ is involutive on points and planes; on a line L = gC, fix k inverting C so that Ň = kC and see that

L ⊥⊥ = (g • kC) ⊥ = (gkC) ⊥ = gk • kC = gC = L.
The polarity is clearly left G-covariant. It also is right-covariant: since I is a normal set, only the case of a line could fail to be clear. But by left-covariance, this reduces to proving conjugacy-covariance, which is obvious. It thus remains to check that the polarity preserves incidence. There are three cases to consider: We wish to make some more comments on the final contradiction.

Remarks.

• One does not need the full strength of Borel's theorem: since C is abelian and the action of PGL4(K) on P 3 (K) is a factor of the natural representation of GL4(K), we may trigonalise simultaneously and find a fixed point for the preimage of C in GL4(K). So all we use geometrically seems to be the 2-nilpotent version of the Lie-Kolchin theorem.

• It is well-known that Hilbert interpretation does not rely on Chevalley-Zilber indecomposable generation (parenthetically said, and contrary to widespread belief, Schur-Zilber field interpretation does not either: see [START_REF] Deloro | Linearisation in model theory[END_REF]), so definable coordinatisation would carry to other model-theoretic settings, say finite-dimensional theories.

• On the other hand, to find a fixed point one does need algebraic closedness of K, which is typical of ℵ1-categorical behaviour. And indeed, real closed fields do not satisfy Borel's theorem; and indeed, SO3(R) does exist.

In the o-minimal case, one may argue in favor of using Bachmann's theorem in order to explicitly identify a real form of PGL2; but not in finite Morley rank in order to prove a contradiction.

Corollaries of Theorem A

We shall be brief. Proof. We first invoke [START_REF] Altınel | Simple groups of finite Morley rank[END_REF]: an infinite simple ranked group of infinite 2-rank is algebraic, hence has non-nilpotent Borel subgroups. As such, we may assume that G is U ⊥ 2 . We are not using this major result out of mere laziness; we actually believe that [ABC08] covers (and exhausts) a topic other than ours.

Corollary

We then refer to the initial analysis in [BN94, Theorem 13.3]; there ads (i)-(iii) (or in the proof, Claims (a)-(e)) are straightforward and do not rely on much. In particular, we shall admit that all Borel subgroups (which here are the definable, connected, proper subgroups maximal as such) are conjugate and partition G. Adding involutions to the picture brings us under the assumptions of Theorem A, and this contradicts the partitioning property.

The B-Sides

Here is the other theorem we announced in the introduction. We remind the reader that hereditarily conjugate means that every definable connected subgroup of G enjoys Borel conjugacy, viz. conjugacy of its maximal definable, connected, soluble subgroups.

Theorem B.

Let G be a connected, U ⊥ 2 , ranked group in which all Borel subgroups are nilpotent and hereditarily conjugate. Then the (possibly trivial) 2-torsion of G is central.

We shall first prove Theorem B, then its Corollaries B.1 and B.2, which will be recalled hereafter.

Proof of Theorem B

Proof. Let G be a counterexample of minimal rank. Notice that any definable, connected subgroup of G, or any quotient by a definable, normal, soluble subgroup of G still satisfies the assumptions (in the latter case, the rank need not be smaller). Claim 1. We may assume that G is centreless.

Proof of Claim. Suppose the theorem is proved for centreless groups.

If Z(G) is finite, then G/Z(G) satisfies all hypotheses and is centreless [BN94, Lemma 6.1]; by our extra assumption, the 2-torsion of G/Z(G) is central, hence trivial as Z(G/Z(G)) = 1: this means that the 2-torsion of G was in Z(G), as desired.

If Z(G) is infinite, then G/Z(G) satisfies all hypotheses but has smaller rank; by induction the 2-torsion of G/Z(G) is central, meaning that the 2-torsion of G is in Z2(G) (the second centre). By [BN94, Theorem 6.9], the (therefore unique) maximal 2-torus

T of G is central in Z2(G), hence characteristic in G, hence central in G [BN94,
= C2. ♦ Claim 3. If x ∈ G is a strongly real element then x 2 ∈ G C g . Proof of Claim. Say x = ij for distinct (possibly commuting) involutions i, j. Let H = C • G (x)
, a definable, connected, proper subgroup. If i inverts H, then let A = H; if not, invoke Reineke's Theorem [BN94, Corollary 6.5] and let A be any infinite, definable, connected, subgroup of C • H (i) > 1. In either case A is an infinite, abelian, x, i -invariant group. We contend that A is contained in a unique conjugate of C.

If we fix one Borel subgroup B containing T , then by nilpotence B ≤ C; by conjugacy, all Borel subgroups are contained in conjugates of C (equality is not accessible a priori, since there are no assumptions on the inner structure of C). But A certainly extends to a Borel subgroup, hence also to a conjugate of C.

Hence we may suppose that A ≤ C; as the latter is ti, it also is x, i -invariant. Now there are two cases.

• Suppose that i inverts C. Let t ∈ T ≤ C and compute:

t x = t -xi = t -ix -1
= t x -1 , so that x 2 centralises t, and x 2 ∈ C(T ) = C.

• Suppose not. As there is a 2-torus Ti containing i, with centraliser Ci, we see that C ∩ Ci = 1, so by disjunction C = Ci. This means that i ∈ T . Bear in mind that x normalises T , so that:

x -1 = x i = ixi = xi x i ∈ xT.
This time x 2 ∈ T ≤ C.

In either case, x 2 lies in C. ♦ Claim 4. If x ∈ G is a strongly real element, then x ∈ G C g .

Proof of Claim. If

x is an involution we are done. We may suppose that 1 = x 2 ∈ C, and in particular x ∈ N . If N/C has odd order then we are done. Now suppose that N/C has even order, so Proposition 1 applies. Write the smallest definable subgroup containing x as x def = z def ⊕ α with x = zα so that z def is a 2-divisible group and α a 2-element ([BN94, Exercise 10 p. 93] if necessary). Observe how z normalises C. Since N/C has order 2, we find z ∈ C. If α ∈ C we are done. Otherwise α ∈ N \ C forces α to be an involution inverting C, hence also z. So x = zα is an involution, which proves Cx = Cα. ♦

Theorem A now provides the desired contradiction. This completes the proof of Theorem B.

Corollaries of Theorem B

Corollary B.1. Let K be a ranked field of characteristic 0 and G ≤ GLn(K) be a simple, definable, non Zariski-closed subgroup. Then G has no involutions.

Proof. Clearly G is U ⊥ 2 . By [Poi01, Théorème 3], Borel subgroups are abelian (hence nilpotent). Moreover, Mustafin [Mus04, Proposition 2.11] proved conjugacy of the Borel subgroups in all definably linear groups in the ranked setting, so the property is hereditary. Hence we are under the assumptions of Theorem B; conclude by simplicity.

Corollary B.2. Let G be a connected ranked group in which all Borel subgroups are good tori. Then the (possibly trivial) 2-torsion of G is central.

Proof. Here again G is necessarily U ⊥ 2 ; moreover, Borel subgroups are maximal good tori, hence abelian and conjugate. And since any definable, connected subgroup of a good torus is still one, the property is hereditary.

Remarks.

• It is unclear to us what would happen to Corollary B.2 with decent tori instead of good tori since hereditary properties are then lost. The situation is even worse with Cartan subgroups (i.e. centralisers of maximal decent tori).

• Despite vague attempts, we could not prove the following.

Conjecture (Borovik-Burdges [BB08, Conjecture 1]).

There is no simple ranked group in which all strongly real elements lie in G C g for C a Cartan subgroup.

This does not seem directly related to the A1-conjecture, which we hope to return to shortly.

This was one of two papers started by the authors in 2018-this is: Paris Album No. 1. Final details were completed during the trimester program "Logic and algorithms in group theory"; hospitality of the Hausdorff Institute in Bonn is warmly acknowledged as we are extremely grateful to the organisers and all staff. We also wish to thank Gregory Cherlin and the anonymous referees for useful comments.

The second author was partially supported on this project by the Sacramento State Research and Creative Activity Faculty Awards Program.

  action); (v) C consists of elements which are strongly real in G (alt.: C ⊆ I • I);

  (i)⇒(ii). Let g ∈ G \ N such that N ∩ N g contains an involution k. If N/C had odd order, then we would obtain k ∈ C ∩ C g , whence C = C g and g ∈ NG(C) = N , a contradiction.

(

  ii)⇒(iii). Suppose that N/C has even order. Lifting torsion, let α ∈ N \ C be a 2-element with least possible order, so that α 2 ∈ C.However α ∈ Cα = C, so α 2 ∈ Cα ∩ C = 1 and α is an involution. Now Cα = C implies that α inverts C. If i ∈ C is any involution then C = C • G (i); moreover i normalises Cα = C so i inverts Cα, which proves that C contains a unique involution. Hence the Prüfer 2-rank is 1, and since α / ∈ C normalises C, the Sylow 2-subgroup of G is as in PGL2(C) (folklore; [DJ10, Proposition 27] for reference).

(

  iii)⇒(iv). Consider a Sylow 2-subgroup, isomorphic to Z/2 ∞ Z Z/2Z, with central involution i ∈ C and some non-central involution k. Every involution in C is toral in C and the Prüfer rank is 1, so i is the only involution in C. In particularC = C k , so k inverts C. Hence N ≤ CG(i) ≤ N (C • G (i))= N , so by Steinberg's torsion theorem[START_REF] Deloro | Steinberg's torsion theorem in the context of groups of finite Morley rank[END_REF], N/C has exponent 2. By the structure of the Sylow 2-subgroup, N = C k .

  with rank rk C -rk D. And the set SC = IC • IC of strongly real elements of C has rank at most 2 rk C -2 rk D. We first estimate rk Σ. If 1 = x = rs ∈ C is strongly real in G (i.e. with r, s ∈ I), then r, s ∈ N so actually r, s ∈ C: hence x ∈ SC . This shows: rk Σ ≤ rk G -rk N + rk SC ≤ rk G + rk C -2 rk D. Now consider the following restriction of the product map:

  point-line: by flag-transitivity and G-covariance, it suffices to prove it for 1 ∈ C; now in usual notation, C ⊥ = N \ C ⊆ I = 1 ⊥ ; theorem); since there are no definable groups of field automorphisms in our ranked context [BN94, Theorem 8.3], one finally finds G ≤ PGL4(K). Consider the Zariski-closure C of C inside PGL4(K); being closed, connected, and soluble (the latter can be seen as an instance of [BN94, Corollary 5.38]), it has a fixed point in its action on the complete variety P 3 (K) [Hum75, Theorem 21.2]. Hence so does C ≤ C: violating regularity. ♦ Theorem A is now proved.

  Theorem 6.16]. So T ≤ Z(G) and G/Z(G) has no non-trivial 2-torus, which by [BBC07, Theorem 1] implies that it actually has no involutions. Here again all the 2-torsion of G was in Z(G).♦Suppose that G contains involutions; let T > 1 be a maximal 2-torus of G, and C = C(T ) be its centraliser, which we know by [AB08, Theorem 1] to be connected. Also let N = N (T ) = N (C(T )) be its normaliser, a finite extension of C by the rigidity of tori[START_REF] Borovik | Groups of finite Morley rank[END_REF] Theorem 6.16]. Claim 2. C is a ti-subgroup of G. Proof of Claim. Let C1, C2 be conjugates of C meeting in x = 1. Then H = C • G (x)is a definable, connected, proper subgroup of G; by induction, its 2-torsion is central. Since it contains both T1 and T2 (the maximal 2-torus of C1, resp. C2), this forces T1 = T2 and C1

  Then G has no involutions.Proof. C is ti and almost self-normalising by a clear rank computation; if G contains involutions then Theorem A applies and G = g∈G C g does not contain all its strongly real elements, a contradiction. Let G be an infinite simple ranked group whose proper, definable, connected subgroups are all nilpotent. Then G has no involutions.

	Corollary A.2.

A.1. Let G be a connected, U ⊥ 2 , ranked group. Suppose that G has a definable, connected subgroup C < G whose conjugates partition G.

point-plane: here we may work with 1 ∈ xI, meaning that x is an involution; hence (xI) ⊥ =

x ∈ I = 1 ⊥ ; line-plane: translating again, suppose C ⊆ xI, forcing x to be an involution inverting C; hence (xI) ⊥ = x ∈ Ň = C ⊥ . ♦ Claim 7. Γ is a projective 3-space.

Proof of Claim. The notion has a variety of equivalent definitions. We adopt the following: a projective 3-space is a transitive point-line-plane incidence structure satisfying the following axioms as given by Hartshorne [START_REF] Hartshorne | Foundations of projective geometry[END_REF].

(S1) Every pair of distinct points lie on a unique line.

(S2) Three non-collinear points lie in a unique plane.

(S3) Every line and plane meet in at least one point.

(S4) Two planes intersect in at least a line.

(S5) There exit four non-coplanar points, no three of which are collinear.

(S6) Every line has at least three points.

First notice the following compatibility property: if a = b are two distinct points of a plane Π and L is some line through a and b, then L ⊆ Π. (This is a consequence of Hartshorne's axioms, but also something we need on our way towards them; notice that we make no claims on uniqueness of L yet.) By flag transitivity, it suffices to deal with b = 1 and a ∈ C \ {1}. Then a plane containing 1 = a must be of the form kI where k is an involution inverting a; notice that even if

We turn to proving the axioms. The last two are trivial; Axiom (S1) is exactly G = G/N C g , since for x = y ∈ G only xC x -1 y will contain both. Axiom (S4) is a weakening of (S1) ⊥ . Axiom (S2) follows from another property.

(T) If a point does not lie on a line, then there is a unique plane containing both.

(Existence in (S2) follows from (T) modulo (S1); uniqueness also requires the "compatibility" above). Finally, Axiom (S3) is a weakening of the dual of (T), which we now prove in our setting.

By flag transitivity, we may suppose that g ∈ G does not lie on L = C. Let Cg = C • G (g) = C and ig = i be the involution in Cg. As we know from Claim 2, there is an involution j commuting to both i and ig and distinct from both. Then j normalises C and Cg but lies in neither, so it inverts both. In particular, g ∈ jI and C ⊆ jI, so the plane jI meets the requirements. We also contend that this plane is unique. If xI contains C (hence 1), then x is an involution, and if xI also contains g, then it contains ig since this lies on the line through 1 and g. Thus, x centralises, i and ig, so uniqueness of the plane amounts to uniqueness of j above, also given by Claim 2. ♦

Remark. The Veblen-Young axioms for a projective space provide an alternative to the (disappointingly asymmetric) axioms of Hartshorne above. To use the former, one needs to show that if a, b, c, d are distinct points for which the line through a and b meets the line through c and d, then the line through a and c meets the line through b and d. Using the above "compatibility" property together with (T), this is rather trivial. Indeed, we may assume a, b, c are noncollinear, so they lie in a plane, which we may take to be I. By assumption, the line through c and d has two points in I, so the entire line, hence d, is in I.

Thus, what we are really showing is that two lines in I must meet, which is just expressing that two distinct involutions commute with a third.

But of course this alternative approach still uses our accurate knowledge of involutions.

Claim 8. Contradiction.

Proof of Claim. G acts regularly by left-translation on the points of Γ. But Γ is a 3-dimensional projective space, hence coordinatisable as Γ P 3 (K) by the "Desargues-Hilbert" theorem ([Hil99, § § 24 sqq.], [Art57, § 2], or [Har67, § 7] for the planar case; bear in mind that every 3-dimensional projective space is arguesian, for instance [Har67, Theorem 2.1]). Now G acts regularly on this, implying G ≤ Aut(P 3 (K)) PGL4(K) Aut(K) [BN94, Fact 3.5]. Of course K is algebraically closed by Macintyre's theorem on fields [BN94, Theorem 8.1] since Desargues field interpretation is definable (something obvious with Hilbert's proof but harder to see with Artin's method-unless one understands Hrushovski's group configuration


Helen Siedel
email: helensiedel@hotmail.com

"First you have to know it exists

Keywords: Mathematics education, resources, curriculum resources, resource system, resource selection

Professional development activity with a secondary school mathematics department in England aimed to further teachers' knowledge of their resource options in order to better inform the resource choices they make and support their ability to use resources to design effective instruction. Participants conceptualized individual resources as representatives of resource types, drafted a model to represent types, and compared types using indices (exponents) as a focal topic. This paper reports on a study of that activity. Findings suggest that learning more about resource options assists and interests teachers and that using a focal topic to explore, characterize, and compare resources has merit. Organizing resources by type for use as a tool for finding resources and coordinating their use in a resource system appears beneficial, but would require work over time.

Introduction

Resource options for teachers are burgeoning. However, despite calls for providing opportunities to develop teachers' resourcing skills (e.g., [START_REF] Rolando | The use of internet tools for teaching and learning by in-service biology teachers: A survey in Brazil[END_REF], and the recognition that teachers choose to use multiple resource types even in countries where prepared curriculum materials such as textbooks have dominated (e.g., [START_REF] Webel | Teachers' evaluations and use of web-based curriculum resources in relation to the Common Core State Standards for Mathematics[END_REF], little is known as yet about how to help mathematics teachers select resources advantageously on behalf of student learning. One question that arises is whether teachers are familiar enough with the full range of available resources to make well-informed selections. In order to select and use a resource, a teacher must first know that it exists, as suggested by the quote from a secondary mathematics teacher in England in the title of this paper. Unknown resources are effectively inaccessible. [START_REF] Siedel | Teachers' selection of resources in an era of plenty: An interview study with secondary mathematics teachers in England[END_REF], in a study of secondary mathematics teachers in England, found that teachers with considerable autonomy and responsibility for resource selection did not know about or were not selecting several potentially useful types of resources. To investigate further, the authors introduced a construct called "the pool of possibilities" (p. 121) to represent all instructional resource options for a teacher population, including human, cultural, social, material and digital resources. Teachers in the study discussed here, the Resource Types study, had an opportunity to learn more about their resource options by exploring their pool of possibilities.

Although the pool of possibilities and the work teachers undertake to find resources in a pool of possibilities have received limited attention in studies of teacher-resource interactions, literature about teachers' selection, adjustment and construction of resources was pertinent to the Resource Types study (e.g., [START_REF] Pepin | Refining teacher design capacity: Mathematics teachers' interactions with digital curriculum resources[END_REF]. One premise for the study was that knowledge of resource options, including familiarity with the range of resource types, supports what is described as teachers' "mathematics-didactical design capacity" (Pepin, Gueudet, & Trouche, 2016, p. 2), that is, their ability to use or construct resources to plan effective instruction. The "focal topic approach" introduced in this paper, for example, illustrates that examining and comparing a variety of resource types for a particular mathematical topic might augment teachers' knowledge of mathematics, of teaching, of student learning, or of resource features. Knowledge of options can thus support the development of other resources, regardless of whether a teacher chooses the options for further use.

The idea of "resource system," an "evolving notion" in mathematics education (Ruthven, 2018, slide 2), framed the study. When teachers interact with various resources in various ways, they typically acquire a collection of individual resources to support their work. As described by [START_REF] Ruthven | The construct of 'resource system' as an analytic tool in understanding the work of teaching[END_REF], a collection of resources becomes a resource system by application of an organizing principle intended to make the collection function more effectively. Often the organizers are teachers, who may seek to unite stand-alone resources in some way to produce a coherent whole, a resource system. Teachers in the Resource Types study were invited to organize resources using the principle that individual resources are representatives of resource types that can be usefully classified and related. Participants applied this principle to a wide range of individual resources and types of resources from their pool of possibilities, not all of which were familiar. The supposition was that identifying types of resources and classifying by types could, first, enable and motivate teachers to select from a wider range of options in their pool of possibilities by increasing the number of resources familiar to and thus accessible to them. Second, regardless of how teachers prefer to use or construct resources, knowing more about a variety of types, and comparing types, might support more advantageous resource selection and ultimately a stronger resource system.

The research question for this study was: What sorts of professional development activities enable secondary mathematics teachers to draw on a wider range of resources when developing their resource system?

Methods

The two-hour Resource Types session occurred in the final month of a nine-month pilot study about the mathematical symbol known as the minus sign, conducted with the mathematics department for 11 to 16 year olds at a state secondary school in the South East Region of England. All eight teachers in the department were present for the Resource Types session. At the inaugural session of the overall pilot, teachers were introduced to instructional resources as a topic of interest for their department that would be folded into the minus sign exploration; they also completed surveys about their resource use. Indices (exponents), the mathematical topic for the Resource Types session, was included in the pilot as one of the topics where the minus sign is known to be problematic.

The Resource Types session had two parts. Part 1 introduced the idea of classifying resources by type. In Part 2, using the "focal topic approach," indices (exponents) was a focal topic for comparing diverse resource types featuring content specific to that topic. Both parts included guided discussion. The full session was audiotaped using two recorders; slides used during the session were posted afterwards. At the department meeting one week after the Resource Types session, participants completed a follow-up survey and had an opportunity to sketch their thinking about a typology for resource types. Participants were encouraged to add comments.

In Part 1, using the "list approach," teachers worked in pairs to brainstorm categories for types of resources and began to organize the types, first using a list of sixteen familiar resources, then a list of twenty-two others, less familiar, and thirdly a proposed list of categories for resource types accompanied by commentary and questions. The department head, also a teacher participant, led discussion after the group had time with each list. Part 1 was open-ended and fast-paced, not intended to lead conclusively to any well-developed classification by types. Part 1 concluded with the group's first attempt to graphically organize "types of resources."

In Part 2, participants reviewed the treatment of indices in a researcher-selected set of resources representing types, in order to examine, characterize, and compare types. The set of types included two contrasting blogs, a research article, traditional curriculum materials, a site to design assessments, a book with rationale for mathematics procedures, a mathematics information site, and a mathematics educator's website with teaching suggestions for numerous topics. These represent some types of resources not mentioned by most teachers in the Siedel and Stylianides study (2018).

Part 2 initially focused on two blog posts by Dan Meyer (dy/dan, 2015; dy/dan, 2016) because these exemplified the value of searching for resources using non-British mathematical terms ("exponents" rather than "indices"), contained rich user discussion that mentioned other resource types, included a "research into practice" example, and included a variety of teaching strategies and activities. Those posts and the research article [START_REF] Cangelosi | The negative sign and exponential expressions: Unveiling students' persistent errors and misconceptions[END_REF] were characterized and compared as types. Due to time constraints, examination of the remaining types was brief.

The follow-up survey was administered at the department meeting a week post-session. Drawing on Guskeys's (2016) levels of evaluation for professional development, which also informed data collection and analysis for the session itself, the survey was designed to determine participants' affective response to the Resource Types session, their learning about individual resources or about the idea of resource types, and whether as individuals or as a department they were likely to use what they learned. At that post-session meeting, as part of a separate activity, teachers were also invited to diagram or otherwise describe their current thinking of how types might relate.

Results

Data were identified by researcher observation of the session, review of the audiotapes, and teachers' written responses in the follow-up activity. Data from conversations in Parts 1 and 2 of the session were qualitative. Teachers appeared fully engaged throughout the session; there was evidence they were learning about individual resources and about resources as exemplars of types. Quantitative survey data (Table 1) and teachers' written comments (Table 2) supported this.

During Part I, resources in the first of three lists participants worked with were familiar to most. They were asked to use the list to generate categories for resources from the new perspective of resources as types. They quickly generated categories that appeared to be based on what a resource facilitates, such as "problem-solving," "worksheets and homework," "develops ideas," "direct teaching." With the second list, however, the resources were less familiar; much of that conversation consisted of teachers trying to learn about a resource in order to identify it by type. For example, someone familiar with a resource could describe it to others. This was an opportunity for participants to learn about the existence of a resource. Categories for types from this list included "technology" and "what level it addresses." These were broader categories, less activity specific. The third list, a proposition of types, surprised teachers because there were many (twenty) types.

As teachers worked with the 3 rd list, teachers remarked about some they had not thought of, such as "books." Later, in Part 2, reviewing types of resources for indices, the group decided to purchase a book that was mentioned; they had not known about the book. Such incidents were evidence that participants were learning about previously unknown resources or broadening their range of types.

During Part 2, participants had an opportunity to evaluate the pre-selected set of individual resources representing a variety of types for the topic indices. Teachers thus learned about these individual resources, while other individual resources were mentioned in the discussion. For example, a discussion about an assessment site prompted mention of a forthcoming resource for UK educators associated with the assessment site. Most participants had not known of its existence.

Results from the follow-up survey, shown in Tables 1 and2, indicate that as a result of the session, some teachers were already visiting online resources in the following week. One teacher, for example, went to a site providing information about mathematics to look up definitions. In the last comment of Table 2 (Teacher 8) the participant expresses an interest in using as wide a range of resources as possible, which would likely involve searching for new resources.

The tables indicate teachers were likely to use ideas about "types of resources" to organize their own interactions with resources or to consider relationships among resource types. The data in Table 1, from the follow-up survey, attest to interest in, ongoing thinking about, and discussion of resource types. Most teachers found the session useful; some had been actively engaged afterwards. Table 2 suggests that some teachers will continue using the teacher-student dichotomy they generated as the basis for a typology (Teacher 2). Comments in response to a question about what stayed with them after the session suggest that these teachers are likely to think about the benefits of classifying resources (Teacher 4), but are aware that there are many types (Teacher 3). Thinking of types promotes learning about their work with resources (Teachers 5 and 8) and, for some, relating them. It also indicates awareness of the difficulty of organizing by type (Teacher 6), with fewer categories a possible objective. Some want to pursue this further (Teacher 7).

Teachers' attempts to graphically organize resources as types also contributed data. Asked to diagram the categories for resource types during the session itself, one pair proposed two unlinked central categories, labeled "teacher resources" and "student resources." The group accepted this. They began to place other categories they generated from the first list as subcategories. At this point the visual resembled a pair of main idea webs. One teacher noted that a resource could also be "mixed," with provisions for both teachers and students. A week later, at the department meeting where participants completed the follow-up survey, but as part of activity apart from the survey, teachers were asked to diagram or discuss their current thinking about how to diagram the relationship of resource types. Five teachers responded. Four used the teacher-student dichotomy, however one of these drew an arrow from teacher to student, linking the two. Another drew three intersecting circles, as in a Venn diagram. These were labeled "teacher," "students," and "online;" directional arrows indicated two-way interaction. The teacher who did not use the teacher-student dichotomy instead began a main idea web with "resource" at the center and two ideas as subcategories. One was labeled "needed for"; the other was not yet labeled. Teacher 2 (Mentioned a site she went to) "I like this site as being a useful resource for teaching but it is not necessarily so for learning."

Discussion

This small study represents a first step towards explicit investigation of a pool of possibilities as a way to enhance teachers' work with instructional resources. Results suggest that opportunities to classify resources by type can amplify teachers' knowledge of their resource options, thus making more resources in their pool of possibilities available to them. Results also point to the complexity of organizing resources by type. More time would have been necessary for participants to fully classify resources and link the classification to the development of a resource system.

With respect to the research question, results seem to imply that the activities developed for the session were responsible for favorable results. However, other factors might account for what occurred with this particular group, who, as colleagues, represent an important resource in their pool of possibilities. These factors include: (1) participant teachers were a team who worked together daily with the same student population, were accustomed to and enjoyed working together on the types of activity in the Resource Types session;

(2) two mathematical topics for the session, indices and the minus sign, were important to this group, and possibly a motivating factor; (3) the larger pilot in which the Resource Types session was embedded featured a monthly topic and accompanying materials that tacitly, but consistently, evidenced the idea that there are a variety of resources to explore for any topic, including resources not specifically for student activity.

Participants were accustomed to exploring multiple types of resources for a topic.

Despite this, participants' reactions to the activities in Parts 1 and 2 suggest that the "list approach" and the "focal topic approach" have the potential to influence teacher thinking in ways that support more deliberate selection from a wider range of resources. During Part 1, participant thinking evolved. With the first list, it seemed straightforward to organize resources by pedagogical activity, but the second list seemed to indicate that resources could be conceptualized various ways, and that organizing by type could be challenging. In response to the third list, of resource types, teachers were not in favor of having too many categories for types, even when, or perhaps especially when, they saw how many types there could be. Part 1 discussion showed that teachers' thinking about the merits of classifying by types varied. Like many mathematics departments in England, this department had not adopted a textbook (e.g., [START_REF] Siedel | Teachers' selection of resources in an era of plenty: An interview study with secondary mathematics teachers in England[END_REF]; guided by the national curriculum, they were responsible for their own curriculum development and resources. One teacher wanted to use as many resource types as possible to guide searches, so that the department would not miss knowing about important ideas, new developments, or any potentially useful resources; the teacher also emphasized the importance of finding "research-based" resources. Another teacher was less interested in broad-based searching by types; this teacher was motivated by "what you do with it" or "what you want it for." In a later session, this teacher defined a high quality resource as one where, "You stop looking! No need to look … again as it does the job perfectly."

As a result of the comparison of types for indices, a participant visited a resource she uses regularly to check its treatment of indices, particularly of zero as a superscript, exemplifying that the "focal topic approach" may influence teachers' work with resources they already have. When one variable that characterizes resources, the mathematical topic, is controlled for, it is easier to compare and comprehend other characteristics, such as the affordances and constraints of each as a type. At the same time, because each resource features the same topic, this activity may augment teachers' knowledge of the topic.

One way in which a typology of resource types for developing a resource system might affect student learning is by possibly reducing the variation in teachers' individual resource collections.

Siedel and Stylianides referred to the existence of variable collections of resources within the same school as "plurality and variation" (2018, p. 130). If teachers' individual sets of resources include multiple resources, with sets varying considerably even among teachers at the same school, there may also be variation in opportunities for children to learn mathematics, known to be problematic in education (e.g., [START_REF] Morris | Creating shared instructional products: An alternative approach to improving teaching[END_REF]. The question of whether or not they should all be using the same resources for a topic was posed to this group, but limited time did not allow discussion.

Two related questions for the mathematics education community emerged from this Resource Types study. The first concerns teachers' understanding of the expression "evidence-based." What types of resource can be considered evidence-based and how can teachers weigh these for resource selection? When comparing blog postings and the Cangelosi et al. (2011) research study, one teacher said the research article was evidence-based but the blog was characterized by opinions.

The implication was that the research article had more worth. Would this mean that ideas developed from experience, such as long-term practice, are less trustworthy? What about the work of influential educators who guide mathematics teachers but may not be conducting systematic research? For their work to be trustworthy is formal review by other educators necessary? Teachers might benefit from opportunities to identify types but also to weight them.

The second, related question deserving attention in studies of teachers' resource skills is how to help mathematics teachers find research articles such as Cangelosi et al. (2011) that contain valuable information for mathematics teachers. Cangelosi et al. (2011) describes misconceptions about indices held by university students. The article includes activity that secondary mathematics teachers could readily implement to forestall similar misconceptions among their own students.

Unfortunately, the inclusion of material in Search Engine Results Pages (SERP) seems to change over time. Cangelosi et al. (2011), for example, at one time appeared in the results of an online search for "negative exponents" and seemed to be openly accessible, but that no longer seems to be the case. Familiarity with the notion of open access articles and pre-publication copies may not aid teachers. Potentially helpful resources may be inaccessible, even when teachers know they are "out there." This becomes significant for questions about whether a resource exists for teachers.

Overall, the results of this study advocate for repeating the activities in a context outside of the pilot study, with more time to pursue the proposition that these activities might strengthen resource systems on behalf of student learning. The results suggest that in the process of classifying types of resources to develop a professional model for improving their resource systems, teachers can extend their knowledge of resource options in ways that benefit all of their work with resources.

(

  root?) "It's easier when resources are in categories and you know where to go for what" Teacher 5 (What took root?) "Made me think about what resources I was using for what reason" Teacher 6 (What took root?) "The difficulty in classifying resources" Teacher 7 (What she planned to do) "Look online for resource categories & how to 'type' them" Teacher 8 What took root?) "Types of resources and how we use them in planning and teaching" Teacher 8 (What took root?)"The need to include all categories including research"

Table 1 : Affirmative responses to post-session survey items (N=8)

 1

	Frequency of "Yes"	Statements to which they were responding
	responses	to	14
	questions
	8			Found the session interesting
	8			Learned something about resources
	8			Thought about resource types later
	6			Found the session useful
	6			Looked for information as a result of something in the session
	5			Thought about doing something different with learners as a result of the session
	5			Noticed resources more explicitly
	4			Discussed the session with colleagues
	4			Went to a resource directly as a result of the session
	4			Noticed the minus sign more explicitly
	3			Learned some mathematics
	3			Noticed indices more explicitly
	2			Learned strategies for teaching
	1			Did something differently with learners as a result of the session

Table 2 : Selection of teacher comments from post-session survey (N=8)

 2


Chongyang Wang
email: chongyang.wang@ens-lyon.fr

Ulises Salinas-Hernández
email: ulisessh@cch.unam.mx

Luc

Luc Trouche

From teacher's naming system of resources to teacher's resource system: Contrasting a Chinese and a Mexican case

Keywords: Curriculum resources, mathematics teachers, resource system, naming system, international comparison

come

Introduction and research questions

In 2018, the Re(s)sources 2018 International Conference1 was hold, dedicated to teachers' resources work analysis with reflections on DAD for the past ten years. A young researcher workshop session was organized (Gitirana et al., 2018, p. 373) specifically for exploring secondary teachers' resource work by analyzing their lexicons in naming and describing their resources and their documentation work (Trouche, to be published). A name is a syntactic entity denoting an object, and "a naming system resembles a restricted database system that infers the object(s) referenced by a name" (Bowman, 1993, p. 795). In addition, considering the views of the teachers, and the influences from their own languages, cultural and institutional contexts, we refer to two cross-cultural comparative projects: the Lexicon Project [START_REF] Clarke | The lexicon project: examining the consequences for international comparative research of pedagogical naming systems from different cultures[END_REF], which investigates the pedagogical naming systems used by different research communities speaking different languages to describe the phenomena of the mathematics classroom; and the project of [START_REF] Remillard | A cross-cultural analysis of the voice of curriculum materials[END_REF], which addresses the influences from cultural contexts and educational traditions.

We situated our study as part of the research program "contrasting naming systems used by teachers in describing their resources and documentation work: towards a deeper analysis of teachers' resource systems" (Trouche, to be published). Holding a similar concern of Lexicon Project on the cultural contexts and educational traditions through cross-cultural analysis, this program involves researchers from eight languages/cultural contexts (Algerian, Brazilian, Chinese, Dutch, French, Mexican, Turkish and Ukrainian) for a better understanding of teachers' resource work through documenting lexicons (naming systems) on resources employed by teachers. Particularly in this study, to deepen the understanding of teachers' resource systems, two contrasting cases (Chinese and Mexican) are analyzed from two research questions: (1) How understanding teachers' resource system through their naming systems? (2) To what extent each case can illuminate the other one?

Conceptual framework

The resource concerned in the project is from [START_REF] Adler | Conceptualizing resources as a theme for teacher education[END_REF], anything with potential "to source again or differently" (p. 207) teachers' activities, "encompassing materials and also all elements intervening 'upstream' of teaching" [START_REF] Gueudet | Collective work with resources: an essential dimension for teacher documentation[END_REF], p. 1003). However in this study we focus more on the curriculum resources from Pepin and Gueudet (online first): all the material resources that are developed and used by teachers and students in their interactions with mathematics in/for teaching and learning, inside and outside the classroom.

This study is situated in the theoretical field of Documentational Approach to Didactics (DAD) and Culture-History Activity Theory (CHAT). DAD is an empirical approach to teachers' work and professional development through a lens of resource. The creative work of mathematics teachers' interactions with resources in their daily work is coined as documentation work (Trouche, Gueudet, & Pepin, online first). As a process of appropriation and transformation of resources, the interaction includes selecting, modifying and creating new resources, by individual or by a group of teachers working together, in-and out-of-class [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF]. During these interactions, teachers develop schemes of usage that are attached to the resources for a same class of situations, and generate a document as outcome (resources + scheme of usage = document). A resource is never isolated, "each resource must be viewed as a part of wider 'set of resources" (ibid, p. 200), and the "set of resources" is named as his/her resource system. Originated from [START_REF] Vygotsky | Thought and language[END_REF], CHAT (Engeström, 2001) emphasizes that the dynamic of consciousness is essentially subjective and shaped historically by social and cultural experiences. The structure and development of human psychological processes emerge through cultural mediation, historical development, practical activity, and the three are interrelated [START_REF] Cole | Cultural psychology: A once and future discipline[END_REF]. CHAT inspires us that individual teacher' documentation work could be situated into the cultural/social/linguistic contexts where they work in. Tools inspired by these two theories will be introduced in the following methodology part.

Methodology

The naming system project adapts the methodology of reflective investigation from DAD, emphasizing the involvement of teachers throughout research with five principles: (1) long-term follow up; (2) in-and out-of-class follow up; (3) broad collection of resources; (4) reflective follow-up; and (5) confronting teacher's views on his/her documentation work (Trouche et al., online first). As a priliminary work, this study mainly takes the fifth principle. To differ the views of teachers from the researchers', we use the methodological tool of Reflective Mapping of the Resource System (RMRS) [START_REF] Wang | Mathematics teachers' expertise in resources work and its development in collectives. A French and a Chinese Cases[END_REF], where the teacher is invited to draw a RMRS by reflecting on his/her resources, and linking them into a structured way with his/her own naming and category of resources. RMRS emphasizes the teacher's views through his/her continuous reflections: the namings of resources and the structure of RMRS are all from teachers, and it could be incomplete with different versions, and improved along with the development of teacher's resource system, and teacher's deeper understanding on it. This is a qualitative study performed through cases study. For feasibility and convenience reasons, we chose the teachers from schools associated to the authors' universities (in China and Mexico), which allows us to do further research with them. The criteria of the cases are: middle career mathematics teachers; from secondary schools in big cities (not from rural areas); with willingness to join our research and improve his/her teaching; "open minded" towards new resources.

Gao graduated in 1993, majored in education management (bachelor), then she started to teach mathematics in middle schools. She was the ex leader of mathematics Teaching Research Group (TRG) in her school, a school-based non-administrational professional group composed by teachers who teach the same discipline, where the leader is usually the most experienced teacher and in charge of organizing teachers' collective teaching research activities [START_REF] Pepin | Developing a deeper understanding of mathematics teaching expertise: Chinese mathematics teachers' resource systems as windows into their work and expertise[END_REF]. She had two classes in grade 8 and 5 lessons for each class per week. In China, curriculum program is national. There are a limited number of textbooks edited under the control of the national government. Teachers cannot decide which textbooks to use, but they have diverse learning-aid materials as supplementary resources to choose.

For the Mexican part, Brenda is a high school mathematics teacher majored in mathematics (bachelor) and Earth sciences (master). During her bachelor study, she attended a mathematics teaching seminar. Then after graduated, she started to work as a volunteer in elementary schools teacher training. Till the interview, she worked as coordinator of mathematics in a primary school. In Mexico, private high schools can use the national curriculum -coordinated by the government-or the curriculum of the autonomous (public) university of the state where the school is located. Brenda works in a private high school, which follows (with some adjustments made by the school) the curriculum of an autonomous university.

Both of Gao and Brenda were interviewed concerning: (1) their working experience, current work and the teaching equipment conditions of their school; (2) their resources used in daily teaching, including the name, the content, the usage, the source, and the way to maintain and organize these resources; (3) an example of resources integration for a lesson preparation about functions (or algebra); (4) a RMRS by reflecting on her/his resources mentioned. The analysis of the interview was made following a common grid: with the interview transcription texts, we selected out the names of the resources most frequently mentioned by the teacher with personal features/preferences, and analyzed them combining the language and culture with the teacher's description on the usages, then situated these resources into the RMRS to infer the structure of teacher's resource system.

Results

This section presents the results of each case.

Selected resources from resource system in the Chinese case of Gao

In the Chinese case of Gao, a most frequently mentioned character is "题(tí)". In Chinese language morphology, word is formed by a combination of characters. In Gao's interview, by combining with other characters, " 题 (t í)" forms into words like "习题(x í tí)" (exercise), "试题(shì t í)" (examination/test question), "例题(lì tí)"(example), "问题(wèn tí)" (question/ problem) etc. In Gao's case, the three selected resources (see in (1) Learning-aid materials (教辅材料, jiào fǔ cái liào) is a kind of printed and bound materials sold in bookstores, to "aid" learning. It generally contains examples, knowledge explanations and exercises. Gao knew the learning-aide material market well due to her experiences working as an author of a learning-aid book some years ago: "The learning-materials also have life-cycle, and have to be updated regularly", so she visited the bookstore almost before new academic year, to see if there is anything new in the materials that she was following. She bought lots of learning-aid materials as her self-owned resources, and she had an awareness of following the trends of the learning-aid material markets, selecting and accumulating the valuable ones.

(2) Exercise note (习题笔记, xí tí bǐ jì) is a personal paper-pencil notes of Gao. She kept such habits for several years: after a whole day's work, she liked to do exercises (e.g. from the learning-aid materials), and keeps the notes of the valuable ones. She also did the category of these exercises according to the contents (e.g. functions) or types (e.g. drawing). The source of the exercises is not limited to the learning-aid materials, but also from Internet. Many big professional websites/forums for mathematics teaching propose their enterprise applications with the popularization of smart mobile phone. Besides these apps, Gao used a social communication application "Wechat", with which she can discuss and share resources (as file or picture) with colleagues, receive articles about exercises explanation, analysis on knowledge, information about exams, or even videos of other teachers' lessons. She kept all the valuable information in her notes.

(3) School-based exercise booklet (校本习题册, xiào běn xí tí cè) is a series booklets developed collectively by the mathematics TRG in her school. As a teacher of grade 8, Gao was in charge of the two booklets in her grade. The booklets are sent to students as homework, consisted mainly with exercises, as Gao explained, "most of them were selected from my exercise notes."

Crossing the three resources, there appears a chain: the learning-aid materials nourish her exercise notes (self-owned resource), which in return "re-source" her exercise booklets (self-developed resource). This evidenced a kind of "resource cycle" in her resource system (Figure 1 left, the red circiles and arrows were marked by the author according to Gao's explanation in the interview). When situating these resources in Gao's RMRS, we find the different roles of resources as "input", "output" and "hub" (evidenced by the arrows). For the inputs, Gao had resources from learning-aid materials, feedbacks from students in their homework; for the output, she developed her self-owned resources into self-developed resources: her personal exercise notes worked as a hub. Gao selected exercises carefully for students, and kept their feedbacks as important resources. This could be referred to her education background (education management): "I pay much attention on students, and try to maintain good relationship with them, they will like mathematics if they like me."

Selected resources from resource system in the Mexican case of Brenda

Three selected resources of the Mexican case were presented in Table 2. (1) Textbook (manual) is the resource that guides the development of the different topics that are addressed throughout the school year. The elaboration of each textbook (for each of the three secondary degrees) is based on the curriculum (another important resource not analyzed in this study)

Naming in

of the largest university in the country and to which the school where the interview was conducted is incorporated. The work of (re)elaboration of the textbooks is directed by a coordinator of mathematics (who is also the teacher of the school). However, teachers are free to suggest and contribute any concerns or observations they may have. Table 2 shows how collaborative work guides the activity of teachers in a particular way: "...we all see the same". In the use of the textbook as a resource, all stages of Brenda's documentation work (review, analysis of resources, adaptation, reorganization, implementation and reflection) are involved. It is also interesting to notice that in Spanish, "manual" refers to something "we have in hands" and something related and available to the usage, whereas quite different with textbooks. The translation makes one lose the meaning-production universe around.

(2) Two resources (diagramas/esquemas) in which Brenda emphasized its use, during the interview, were the use of diagrams/schemes. It is important to note that both words (diagram and scheme) were used by indistinctly to refer to the same resource: diagram (Figure 2). However, the term "scheme" is incorporated to show how the teachers name their resources in relation to their teaching activities.

Brenda elaborated the diagrams according to her teaching contents. The diagramas were initially used as a way of learning by Brenda when she was a student. This stage is closely related to the analysis of the textbook (manual), as she pointed out: "I make diagrams to see what my objective is and what they [students] need to know for reaching that goal" (see in Table 2). Then, during the lessons, Brenda incorporated the resource as learning material for the students: "I give them copies of the scheme, or blank ones to be filled." (3) School calendar (Calendario escolar). In relation to the resources selected from the interview, the use of the textbook and the elaboration of diagrams are closely related to the teaching plan and lesson design. This includes both the textbook that will be addressed in each class and the elaboration of diagrams for each topic. These two resources are organized from a school calendar.

Crossing the three resources within her RMRS (Figure 2), they are all resources for her "lesson". Brenda did not draw her RMRS centered on "resource" like the Chinese case. However, considering the particularity of the textbook she used, we can still infer that the textbook bridges Brenda's individual documentation work with her collective work with others: The textbook is developed based on the university curriculum with the involvement of the school teachers, and as a teacher, Benda worked as both user and usage feedbacks provider; To implement the curriculum and textbook, she adapted her personal learning strategy (diagram) into classroom teaching, and diffused it with other teachers through resources related to school calendar (teaching plan).

Discussion and perspectives

Resource system is not easy to be inferred from or interviews. In this paper, we took a method of combining teacher's naming system and her/his RMRS, with two contrasting cases from different cultural contexts, China and Mexico, to have more elements (theoretical and methodological) to analyze teacher's resource system. Far more work need to be done for a deeper understanding on their resource systems. Although Gao and Brenda are two very different cases -in terms of their teaching activity and their sociocultural context-, one from public middle school and using national curriculum, and one from private high school using a university curriculum. However, by documenting the name/content/usage as well as their positions in RMRS, we can still find some features that allow us to develop and deepen our further analysis works.

Related to research question (1): In the Chinese case, the resource naming system showed an emphasis on "tí(exercise in English)", and the three selected resources evidenced its position of central element in the teacher's resource system, linking both the teacher's personal working habits (taking notes) and the cultural context (exams-emphasized culture and huge learning-aid material markets) of mathematics education. In the Mexican case, the resource naming system evoked an adaption of personal choices (e.g. diagram adapted from learning teaching) and consideration on others' work (e.g. school calendar), mediated by a collective used/developped/revised manuel (textbook). For research question (2): Crossing the two cases, they are both following a fixed curriculum (national curriculum and local unified textbook in Chinese case, unified curriculum and textbooks produced by the university in Mexican case); they are both trying to integrated the supportive resources (learning-aid materials in Chinese case, disciplinary academic books in Mexican case); they are both organizing their resource work centered on students (selecting exercises from diverse sources and paying attention on their feedbacks in the Chinese case, adapting her own learning strategy, diagram, into classroom teaching in the Mexican case). Additionally, there are school-based resources involving the two teachers in both two cases: the school-based exercises booklets collectively developed by the mathematics TRG in the Chinese case, and the university-curriculum based textbooks developed and revised by the school mathematics teachers in the Mexican case. In the case of Gao, the different terms related to "tí(exercise) reveal a potential resource category from the teacher's view. In the case of the Mexican teacher, the ambiguity of naming the same resource in two different ways infers a more individual work of the teacher.

To summarize, analyzing teacher' naming systems on resources by actuating it within specific cultural contexts could be a promising method for us researchers to deepen the understanding on teacher's resource system. Due to the limited space, we were not able to analyze the mathemtics components of their resource system. As a beginning of this exploration, we hope that involving other cases and crossing with other projects will help us to go further in this direction.

Figure 1 :

 1 Figure 1: Part of RMRS of Gao (original left, and digital transposition by the researcher right)

Figure 2 :

 2 Figure 2: Part of RMRS of Brenda (original left, and digital transposition by the researcher right)

Table 1

 1 ) are the sources of the "题(tí)".

	Naming in	Translated	Description in English	Transcriptions examples
	Chinese	name in
		English
	教辅材料	Learning-a	The commercial textual materials (e.g.	"Before the new semester, I
	(jiào fǔ c ái liào)	id materials	books) for students, with exercises or exam questions or knowledge explanations (available in bookstore).	will go (to the bookstore) to check the new learning-aid materials."
	习题笔记	Exercise	The personal notes of Gao, in which	"I keep notes when I find
	(xí tí bǐ jì)	note	she used for collecting the exercises she	valuable exercises, I made
			considered as valuable.	four (notebooks) last year."
	校本习题	School-bas	Booklet with exercises for the students,	"This school-based exercise
	册 (xiào bě n xí tí cè)	ed exercise booklet	developed collectively by all the mathematics teachers inside Gao's school (not for sell)	booklet was developed by me...which selected from my exercise notes."

Table 1 : Three typical expressions on resource naming by Gao

 1

Table 2 : Three typical expressions on resource naming by Brenda

 2

https://resources-2018.sciencesconf.org