Abderrahman Iggidr
email: abderrahman.iggidr@inria.fr

Max O Souza
email: maxsouza@id.uff.br

On the limits of the Volterra function in the Lyapunov method: the Anderson-May-Gupta model as a cautionary example

Keywords: Lyapunov functions, Global stability, Malaria intra-host model, Anderson-May-Gupta model

The Volterra-type Lyapunov functions are an ubiquitous tool for establishing global stability in systems appearing in mathematical biology. We show, however, that no function of this type can be a Lyapunov function for the endemic equilibria of a classical intra-host model of malaria -the AMG model. More precisely, we give a sharp condition on the model parameters for this to be the case. This condition leaves out a large and biologically meaningful parameter range that will have to be addressed by a different method. We also present a set of three alternative arguments that enlarge the range of parameters for which global stability can be obtained -including parameter ranges that are relevant to malaria.

1. Introduction 1.1. Background. The traditional mathematical analysis of epidemiological models usually starts with basic questions as well-posedness of the model, determination of the basic reproductive number, identification of equilibria, and local stability. Once this is settled, the next step is to establish global stability of the appropriate equilibria in the different parameter regimes. The importance of this analysis can been read from the introduction of [START_REF] Jacquez | Qualitative theory of compartmental systems[END_REF] -see also [START_REF] Fall | Epidemiological models and Lyapunov functions[END_REF]. Indeed, in many models the Sharp Threshold Property holds: (i) there exists an equilibrium without the disease -the so-called disease free equilibrium (DFE) -which exists for all valid parameter values; (ii) the DFE equilibrium is globally asymptotically stable when R 0 ≤ 1; (iii) when R 0 > 1 there exists a unique equilibrium where the disease is present -the so-called endemic equilibrium (EE) -which is then globally asymptotically stable. The rationale behind this property is that although a system might have a complex transient dynamics, its long term behaviour is quite simple, with a dichotomy completely governed by R 0 . See [START_REF] Shuai | Global stability of infectious disease models using lyapunov functions[END_REF] for the original reference and further discussion of this property.

In order to verify the Sharp Threshold Property, one has to check that conditions (i)-(iii) above hold. Condition (i) is typically a straightforward computation. In order to check condition (ii), a linear Lyapunov function is usually sufficient. Condition (iii), however, is typically much harder to verify. In more complex models even existence and uniqueness can be a difficult issue. Indeed, in some cases existence can be ascertained by means of uniformly persistence and then verifying that an interior equilibrium is necessarily globally asymptotically stable yields also uniqueness and check that (iii) holds -eg. [START_REF] Guo | Global stability of the endemic equilibrium of multigroup sir epidemic models[END_REF].

In any case, the use of Volterra type Lyapunov functions has proved to be quite an effective tool to establish that the endemic equilibrium is globally asymptotically stable. This function was first used by Volterra in [START_REF] Volterra | Leçons sur la théorie mathématique de la lutte pour la vie[END_REF], and later by Goh [START_REF] Goh | Global stability in a class of prey-predator models[END_REF] (also [START_REF] Goh | Global stability in many-species systems[END_REF])). It has been rediscovered later on and used for epidemic models by Lin and So [START_REF] Lin | Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations[END_REF] and Korobeinikov and Maini [START_REF] Korobeinikov | A lyapunov function and global properties for sir and seir epidemiological models with nonlinear incidence[END_REF]. Since then, it has been successfully used in innumerous situations -eg. [START_REF] Guo | Global dynamics of a staged progression model for infectious diseases[END_REF] , [START_REF] Fall | Epidemiological models and Lyapunov functions[END_REF], [START_REF] Iggidr | Global analysis of new malaria intrahost models with a competitive exclusion principle[END_REF], [START_REF] Guo | A Graph-Theoretic Approach to the Method of Global Lyapunov Functions[END_REF], [START_REF] Souza | Global stability for a class of virus models with cytotoxic t lymphocyte immune response and antigenic variation[END_REF], [START_REF] Smith | Chemostats and epidemics: competition for nutrients/hosts[END_REF], [START_REF] Thieme | Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators[END_REF], [START_REF] Souza | Multiscale analysis for a vector-borne epidemic model[END_REF].

On one hand, while it is known that, when an equilibrium is globally asymptotically stable, then a Lyapunov function must exist ( [START_REF] Massera | On liapounoff's conditions of stability[END_REF] and [START_REF] Massera | Contributions to stability theory[END_REF]) -the proofs of these results are non constructive and there is no guarantee that it will belong to a particular class of functions. On the other hand, the success of the Volterra Lyapunov function seems to have led to the belief that, at least for biological models, if the model is globally stable, then one should be able to tune the parameters of a Volterra Lyapunov function to show its global stability.

Finally, we would like to mention a few alternative approaches that have been used to show globally stability: (i) Beretta and Capasso [START_REF] Beretta | On the general structure of epidemic systems. Global asymptotic stability[END_REF] use of a skew-symmetry condition on the Jacobian of the matrix of the system to give a necessary condition for the global stability of the endemic equilibrium; (ii) Li and Muldowney [START_REF] Li | A geometric approach to global-stability problems[END_REF] developed an approach that has ben widely used to deal with the global stability of the EE for many epidemic models; (iii) the use of techniques of monotone dynamical systems [START_REF] Hirsch | Monotone dynamical systems[END_REF].

Here we will revisit the stability of a classical intra-host model of Malaria and show that (i) a Volterra-Lyapunov function for this model has to have a very specific form; (ii) this specific form cannot be a Lyapunov function for all biologically feasible parameter values; (iii) the use of three different techniques, namely monotone dynamical systems, asymptotically 2-D systems, and singular perturbations can be used to expand the range of parameters where global stability can be verified. In this particular case, these techniques will include parameter regimes that are relevant to Malaria. We also point out that, to the best of our knowledge, no rigorous proof that the Volterra-Lyapunov function can fail in situations where the model is globally stable has been provided so far.

1.2. The Anderson-May-Gupta (AMG) model. The original AMG [START_REF] Anderson | Non-linear phenomena in host-parasite interactions[END_REF] model is given by (1)

   ẋ = Λ -βmx -µ x x, ẏ = βmx -µ y y, ṁ = rµ y y -βmx -µ m m,
where x is the concentration of uninfected erythrocytes (red blood cells) in the blood, y is the concentration of infected erythrocytes, and m the concentration of free merozoites. The parameters µ x , µ y and µ m are respectively the death rates of the uninfected erythrocytes, infected parasites and free merozoites. The parameter β is the contact rate between erythrocytes and merozoites, it is the rate at which merozoites invade erythrocytes. Death of an infected erythrocyte results in the release of an average number of r merozoites. Λ is the rate at which new red cells are formed.

All the parameters are positive and the state variables x, y and m are nonnegative, i.e., (x, y, m) ∈ R 3 + . The term -βmx in the third equation is important from a modelling point of view, for parasite and viral infections -in this case, differently form contact infections described by models like SIR or SEIR, the infective agent must enter the cell, and it is not able to infect other cells directly. However, its presence complicates the analysis considerably. Modified HIV models containing this term were studied by [START_REF] Leenheer | Virus dynamics: A global analysis[END_REF], but only partial results on the global stability were obtained. If this term is omitted, then System (1) becomes isomorphic to the standard SEIR epidemiological model or the May-Nowak in vivo HIV model [START_REF] Nowak | Virus dynamics: mathematical principles of immunology and virology[END_REF]. For these models, the Sharp Threshold Property holds -cf. [START_REF] Li | Global stability for the seir model in epidemiology[END_REF] and [START_REF] Korobeinikov | A lyapunov function and global properties for sir and seir epidemiological models with nonlinear incidence[END_REF].

We will now briefly review the dynamical properties of the AMG system. If we write x * = Λ µx , then the basic reproduction number is given by

R 0 = rβx * µ m + βx * .
If R 0 ≤ 1 then the only non-negative equilibria is (x * , 0, 0) -known as the disease free equilibrium or DFE. In this case the DFE is both locally and globally asymptotically stablecf. [START_REF] Iggidr | Global analysis of new malaria intrahost models with a competitive exclusion principle[END_REF].

If R 0 > 1 then the system is uniformly persistent [START_REF] Leenheer | Virus dynamics: A global analysis[END_REF] and there exists a unique positive equilibrium -known as the endemic equilibrium or EE-that is given by

x = µ m β(r -1) , ȳ = -µ m µ x + (r -1)Λβ β(r -1)µ y , m = -µ m µ x + (r -1)Λβ βµ m . Note that R 0 > 1 implies r > 1.
The endemic equilibrium is always locally asymptotically stable when it exists -cf. [START_REF] Anderson | Non-linear phenomena in host-parasite interactions[END_REF][START_REF] Iggidr | Global analysis of new malaria intrahost models with a competitive exclusion principle[END_REF]. System (1) would satisfy the Sharp Threshold property [START_REF] Shuai | Global stability of infectious disease models using lyapunov functions[END_REF] if one could show that the endemic equilibrium is globally asymptotically stable, when R 0 > 1. While there are no contrary evidences to this fact, its proof has been elusive. Indeed, in [START_REF] Iggidr | Global analysis of new malaria intrahost models with a competitive exclusion principle[END_REF], partial results are obtained using a Lyapunov belonging to the class

(2) V (x, y, m) = a(x -x log x) + b(y -ȳ log y) + c(m -m log m),
where the coefficients in (2) are chosen such that the linear terms y and m and the bilinear terms in xm that appear in V cancel out. We will denote the function obtained with this particular choice of coefficients as V EE -following the notation used in [START_REF] Iggidr | Global analysis of new malaria intrahost models with a competitive exclusion principle[END_REF]:

(3)

V EE (x, y, m) = (r -1)x( x x -log x x ) + rȳ( y ȳ -log y ȳ ) + m( m m -log m m ).
There are also a number of incorrect or unusable claims in the literature: In [START_REF] Tumwiine | On global stability of the intra-host dynamics of malaria and the immune system[END_REF], the authors claim to prove the global asymptotic stability of the endemic equilibrium using a result of [START_REF] Muldowney | Compound matrices and ordinary differential equations[END_REF] but their proof is not correct since in equation ( 20) they use the second additive compound matrix computed at the endemic equilibrium whereas (according to [START_REF] Muldowney | Compound matrices and ordinary differential equations[END_REF]) the second additive compound matrix must be computed at an arbitrary point. In [START_REF] Niger | Immune response and imperfect vaccine in malaria dynamics[END_REF] (theorem 4), globally asymptotic stability of the endemic equilibrium is addressed under the supplementary condition (1 -x x )(1m m ) ≥ 0 which is not possible to verify.

Main result and outline.

In what follows, the expression Lyapunov function is to be understood as a Lyapunov function for the Endemic Equilibrium of System (1).

We will prove the following results

Theorem 1.1. Let V be candidate Lyapunov function with the form given by Equation (2). Then V is a Lyapunov function for some non-empty subset of model parameters if, and only if, it is a multiple of V EE .

Theorem 1.2. The function V EE is a Lyapunov function if, and only if, we have 4) is sharp, and it improves earlier results: In [START_REF] Iggidr | Global analysis of new malaria intrahost models with a competitive exclusion principle[END_REF], the first sufficient condition is βΛ µxµm ≤ r r-1 = 1 + 1 r-1 , and a weaker condition is βΛ µxµm ≤ (

(4) βΛ µ x µ m ≤ 4 r -1 r - 1 2 2 = 4r + 1 r -1 . Remark 1.1. Condition (
√ r + √ r -1) 2 . In [4]: βΛ ≤ min(µ x , µ y )µ y .
This existence of the sharp condition [START_REF] Leenheer | Virus dynamics: A global analysis[END_REF] shows that the Volterra integral cannot be used to show global stability of the endemic equilibrium for the entire relevant range of parameters. Thus, alternative approaches are necessary and we present three such alternatives that yield partial answers: Firstly, we show that, when µ x = µ y , then the AMG system is asymptotically equivalent to a 2-D system, and then a global stability result can be obtained. Secondly, when µ y < µ x , we show that the AMG system is equivalent to a 3-D cooperative and irreducible system -and once again global stability follows. Finally, when µ y µ x 1 or µ m µ x 1, we present a separation of scales argument that reduces the problem to a 2-D one, where global stability can be shown. The outline of the paper is as follows: Section 2 introduces a rescaled and rewritten equivalent system. Section 3 gives the proofs of Theorems 1.1 and 1.2. Section 4 presents a set of alternative approaches, as discussed above, that show global stability of the endemic equilibrium for a range of parameters which include values not covered by condition [START_REF] Leenheer | Virus dynamics: A global analysis[END_REF]. A conjecture about the global stability of endemic equilibrium for all biologically feasible parameters is also given. A discussion is presented in Section 5.

Basic set-up

A rescaled version of the AMG model.

In what follows, it will be convenient to work with a rescaled version of the AMG model. We rescale time with µ x , and let

X s = x * x = (r -1) βΛ µ x µ m , µ = µ y µ x , ν = µ m µ x .
In addition, let x = xX, y = ȳY, m = mM.

Then the original AMG system can then be written as ( 5)

     Ẋ = X s -X -(X s -1)M X, Ẏ = µ (M X -Y ) , Ṁ = ν rY r-1 -M X r-1 -M .
We also introduce a new parameter g:

(6) g = r(X s -1) X s (r -1) = r 2 (R 0 -1) (r -1) 2 R 0 = r r -1 - r µ x µ m (r -1) 2 βΛ .
From the above, we can write

R 0 = rX s X s + r -1 .
The definitions given above are not completely independent -the relevant equivalences are given in the following result:

Lemma 2.1. The above definitions imply the following equivalences:

(1) R 0 > 1 is equivalent to g > 0, r > 1 and X s > 1.

(2) The following conditions are equivalent:

• βΛ µ x µ m ≤ 4 r -1 r - 1 2 2 = 4r + 1 r -1 (Condition (4)). • (7) X s ≤ 4 r - 1 2 2 . • (8) g ≤ 2r 2r -1 2 .
For System (5) the family of candidate Lyapunov functions given in ( 2) is given by ( 9)

V (X, Y, M ) = A(X -log X) + B(Y -log Y ) + C(M -log M ), A, B, C > 0.
In addition, the function V EE is now given by (10)

V EE (X, Y, M ) = (r -1)x(X -log X) + rȳ(Y -log Y ) + m(M -log M ).
Finally, we will denote by K a compact attracting set for System (5).

2.2.

A convenient set of alternative variables. As it will transpire in the proofs below, much of the stability results will depend on the ratio Y /M , and it turns out that it will be advantageous to set Y = T M and work with ( 11)

               Ẋ = -X + X s -(X s -1) XY T , Ẏ = µ X T -1 Y, Ṫ = µ(λ -1)T + µX - λµr r -1 T 2 + λµ r -1 T X, with λ = ν µ .
3. Proof of Theorems 1.1 and 1.2

3.1. Proof of Theorem 1.1. Sufficiency follows from the results in [START_REF] Iggidr | Global analysis of new malaria intrahost models with a competitive exclusion principle[END_REF] on the global stability of the Endemic Equilibrium when R 0 > 1. Indeed, the authors proved that V EE is a Lyapunov function provided that the parameters satisfy βΛ µxµm ≤ ( √ r + √ r -1) 2 . To prove necessity, we compute the derivative of V along the solutions of ( 5):

V = - A(X -1) ((M X -1)X s -M X + X) X + Bµ(Y -1)(M X -Y ) Y - Cν(M -1)(M (r + X -1) -rY ) M (r -1) .
A straightforward computation shows that, irrespective of the value of the coefficients A, B and C, one always has that V (1, Y, Y ) = 0.

In other words, V always vanishes in the line X = 1 and Y = M -we will refer to this line as the critical line.

We now study the behaviour of V in the neighbourhood of this critical line:

V (1 + δ, Y, Y ) = ∂ X V (1, Y, Y )δ + 1 2 ∂ 2 X V (1, Y, Y )δ 2 + O δ 3 = (Y -1) 1 -r (X s -1) (r -1) A + (1 -r) Bµ + Cν δ -AX s δ 2 + O δ 3 , and V (1, Y, Y + δ) = ∂ M V (1, Y, Y )δ + 1 2 ∂ 2 M V (1, Y, Y )δ 2 + O δ 3 = δ(Y -1)(Bµ(r -1) -Cνr) (r -1)Y - Cδ 2 νr (r -1)Y 2 + O δ 3 .
From these computations we observe that, unless the terms with order one vanish, there is always a neighbourhood of the critical line (by choosing the appropriate sign for δ) such that we have V > 0. This observation leads to the following necessary conditions to ensure that V ≤ 0:

(12) (X s -1) (r -1) A + (1 -r) Bµ + Cν = 0, Bµ(r -1) -Cνr = 0.
The solutions to the system above can be written in terms of B as

A = B µ(r -1) r (X s -1) , C = B µ(r -1) νr .
With the particular choice

B = rΛ (X s -1) µX s ,
we obtain, using the definition of X s and the equilibrium relations, that

B = µ x rȳ, A = (r -1) Λ X s = xΛ x * (r -1) = µ x (r -1)
x, and C = µ(r -1) νr µ x rȳ = µ x m. Hence,we arrive at

V = µ x V EE , which completes the proof. 3.2. Proof of Theorem 1.2.
In this proof, it will be convenient to work with V = V EE Λ(r-1) and write V using the parameter g. In this case, we obtain

(13) V = 2 - 1 X -X + g 1 + X - Y M - M X Y .
On noticing that V depends only on X and on the ratio between Y and M , we opt to work in the projective line of the Y -M plane. Thus, we write Y = T M , and work in the alternative variable setting described in Section 2.2. In this setting, we have

V = 2 - 1 X -X + g 1 + X -T - X T .
Remark 3.1. In this new set of variables the critical line becomes X = 1 and T = 1.

Thus V = 0 is equivalent to the following cubic plane algebraic curve:

(14) p(X, T ) = X 2 (T (g -1) -g) + T X(g + 2 -gT ) -T = 0.
For g = 1, we can write the solution to p(X, T ) = 0 for fixed T as

X ± (T ) = g 2(g -1) T (T -T + ) ± (T -1) T (T -T c ) (T -T * ) ,
where

T * = g g -1 , T c = 4 g and T + = g + 2 g .
It is also convenient to write the product and sum of the roots:

Π(T ) = -1 g -1 T T -T * and Σ(T ) = g g -1 T -T + T -T * T .
Notice that we can use the expressions above when g = 1, if we replace (g -1)(T -T * ) by 1.

The following list of facts is easily checked using the expressions above for p, X ± , Π and Σ:

(1) p(1, 1) = 0 for any value of g and, in particular, X ± (1) = 1 -this will be termed the trivial solution;

(2) X ± (T ) is real if, and only if, T ≥ T c . In addition, X ± (T c ) = 2 g -2 for g = 2;

(3) When 0 < g ≤ 1 the only positive solution is the trivial one;

(4) When g > 1, we have

X + (T ) = g (g -2) (-1 + signum(g -2)) 2 (g -1) 3 1 (T -T * ) + g 3 -3g 2 + 2g + 2 signum(g -2) + g 3 -3g 2 + 4g -4 2 (g -1) 2 (g -2) + g 5 -6g 4 + 14g 3 -18g 2 + 12g -2 signum(g -2) + g 2 (g -2) 3 (2g -2) (g -2) 3 g (T -T * ) +o(T -T * ) .
So in a neighborhood of T * ,

X + (T ) ∼        g (g -2) 2 (g -1) 3 1 (T -T * ) for 1 < g ≤ 2, g -1 g -2 for g > 2.
X

-(T ) = - g (signum(g -2) + 1) (g -2) 2 (g -1) 3 1 (T -T * ) + -g 3 + 3g 2 -2g -2 signum(g -2) + g 3 -3g 2 + 4g -4 2 (g -1) 2 (g -2) + -g 5 + 6g 4 -14g 3 + 18g 2 -12g + 2 signum(g -2) + g 2 (g -2) 3 2 (g -1) (g -2) 3 g (T -T * ) +o(T -T * ) .
So in a neighborhood of T * ,

X -(T ) ∼      g -1 g -2 for 1 < g ≤ 2, -g(g -2) (g -1) 3 1 (T -T * )
for g > 2.

The following hold: (a) we have T c ≤ T * with equality only for g = 2. (b) for T > T * , we have X + (T ) > 0 and X -(T ) < 0. (c) lim

T →+∞ X + (T ) = +∞. (d) lim T →T + * X + (T ) =    +∞, 1 < g ≤ 2, g -1 g -2 , g > 2.
(5) When g > 2, the following hold:

(a) X ± (T ) > 0 for T ∈ (T c , T * ). (b) lim T →T - * X -(T ) = +∞; (c) lim T →T - * X + (T ) = g -1 g -2 .
(6) When 2 < g ≤ 4, X -(T ) ≥ X + (T ) for T ∈ (T c , T * ) and with equality only at T = T c . (7) When 1 < g ≤ 4, the non-trivial solution with minimal X is given by (X m , T m ), where

T m = 1 √ g -1 and X m = X + (T m ) = T 2 m .
(8) When g > 4, the non-trivial solution with minimal X, (X m , T m ), satisfies T c < T m < 1 < T * and X m < 2 g -2 .

In order to proceed with the proof of Theorem 1.2, we note that, in the (T, X) variables, the attracting compact set K is mapped into T ≥ 0 and 0 ≤ X ≤ X s . Thus V will be a weak Lyapunov function if, and only if, the intersection of the zero set of p(X, T ) with K is only the trivial solution.

The proof of the Theorem is completed thanks to the three following facts:

(I) When 0 < g ≤ 1 there is no positive solution other than the trivial one. (II) When 1 < g < 4, there are no non-trivial solutions belonging to K if, and only if,

X s ≤ 1 ( √ g -1) 2 .
(III) When g ≥ 4, then there are always non-trivial solutions belonging to K. Indeed, in this case X ± always intersect transversally at (1, 1) ∈ K and V EE can never be a Lyapunov function, since X s > 1.

These facts follow straightforwardly from the facts listed before; more precisely, (I) follows from ( 3), (II) follows from ( 7) and (III) folows from [START_REF] Guo | Global dynamics of a staged progression model for infectious diseases[END_REF]. See Figure 1 for a few examples of the curves X ± for different values of g.

3.

3. An alternative proof to Theorem 1.2. We will have to distinguish the cases g ≤ 1 and g > 1. When R 0 > 1, we have X s > 1 and it is easy to see that the set {(X, Y, T ) : 0 ≤ X ≤ X s } is a positively invariant set for System [START_REF] Hetzel | The within-host cellular dynamics of bloodstage malaria: theoretical and experimental studies[END_REF]. 8) is satisfied and that f (X) ≤ 0. Now, for g > 1 it is possible to write f (

Recall that we have V (X, T ) = 2 - 1 X -X + g 1 + X -T - X T . 1 + X -T - X T = 1 + X -2 √ X -(T + X T -2 √ X) = (1 - √ X) 2 - (T - √ X) 2 T ≤ (1 - √ X) 2 Hence, V ≤ - (X -1) 2 X + g(1 - √ X) 2 = f (X). If g ≤ 1 then it is clear that Condi- tion (
X) = 1 X (g -1) √ X -1 2 √ X - 1 √ g -1 1 √ g + 1 + √ X . Since g > 1, the sign of f (X) for X > 0 is given by the sign of √ X - 1 √ g -1 . Hence, for X > 0, f (X) ≤ 0 ⇐⇒ X ≤ 1 √ g -1 2
. Therefore, f ≤ 0 on (0, X s ] if, and only if,

X s ≤ 1 √ g -1 2 . We then deduce that X s ≤ 1 √ g -1 2 =⇒ V ≤ 0. On the other hand V (X, √ X) = - (X -1) 2 X + g(1 - √ X) 2 = (1 - √ X) 2 g - (1 + √ X) 2 X = (1 - √ X) 2 g -1 + 1 √ X 2 . Thus V ≤ 0 =⇒ √ g ≤ 1 + 1 √ X
, and in particular

√ g ≤ 1 + 1 √ X S which implies X s ≤ 1 √ g -1 2 . Hence V ≤ 0 ⇐⇒ X s ≤ 1 √ g -1 2 .
We have

X S = r r + g(1 -r) , so X s ≤ 1 √ g -1 2 ⇐⇒ g ≤ 2r 2r -1 2 . Using the ex- pression of g = r(X S -1) X S (r -1)
and the fact that r > 1, the later inequality is equivalent to

r - 4r 2 (2r -1) 2 (r -1) X S ≤ r, which is equivalent to X s ≤ (2r -1) 2 . To conclude, V ≤ 0 ⇐⇒ g ≤ 2r 2r -1 2 ⇐⇒ X s ≤ (2r -1) 2 .

Beyond the negative-definite approach

We will now present alternative arguments that allow to show that System (1) is globally asymptotically stable when µ ≤ 1 and when µ 1 -this will include parameter regimes for which condition ( 8) is not satisfied. When µ = 1 we show that system is asymptotically a 2-D one and for µ < 1 we show that it is equivalent to a cooperative system in an appropriate new set of state variables. , we employ a separation of scales argument (justified by Tikhonov's theorem) for µ 1 which, once again, reduces the problem to a two dimensional one. Finally, we give an example that is not covered by any of these or previous results.

α = µ r -1 ν .
Then system ( 5) is given by ( 15)

       Ṅ = αX s -N -(µ -1)Y, Ṡ = αX s -νS + (ν -1)N + (ν + µ(r -2) + 1)Y, Ẏ = µ 1 αα (S -N -Y )(N -Y ) -Y .
The endemic equilibrium is now written as N = α + 1, S = α + α + 2, Ȳ = 1. Notice that we always have N > Y , S > N and S > 2Y .

If µ = 1, we have ( 16)

     Ṅ = αX s -N, Ṡ = αX s -νS + (ν -1)N + (ν + r -1)Y, Ẏ = 1 αα (S -N -Y )(N -Y ) -Y.
System ( 16) is a triangular system (the first equation depends only on the variable N ), and we have that lim t→∞ N (t) = αX s . Since µ = 1, we have αX s = α( 1 α + 1) = α + 1 = N . All the solutions are bounded. Therefore, the stability properties of system ( 16) are the same as those of the following planar system (thanks to [START_REF] Vidyasagar | Decomposition techniques for large-scale systems with nonadditive interactions: Stability and stabilizability[END_REF], [START_REF] Thieme | Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations[END_REF], [START_REF] Castillo-Chavez | Asymptotically autonomous epidemic models[END_REF]):

(17)

   Ṡ = αX s -νS + (ν -1) N + (ν + r -1)Y = ν( N -S) + (ν + r -1)Y, Ẏ = 1 αα (S -N -Y )( N -Y ) -Y.
Applying Bendixson's Criterion to [START_REF] Korobeinikov | A lyapunov function and global properties for sir and seir epidemiological models with nonlinear incidence[END_REF] we obtain

-ν - S -2Y αα -1 < 0,
since S > 2Y . Therefore, the endemic equilibrium is a globally asymptotically stable equilibrium for System [START_REF] Jacquez | Qualitative theory of compartmental systems[END_REF].

       Ṅ = αX s -N -(µ -1)Y, Ż = -νZ + (µ(r -1) + ν)Y, Ẏ = µ 1 αα (Z -Y )(N -Y ) -Y .
The Jacobian of ( 18) is

J =     -1 0 1 -µ 0 -ν µ(r -1) + ν µ(Z -Y ) αα µ(N -Y ) αα -µ N + Z -2Y αα + 1     .
Since we have Z > Y and N > Y , then J is Metzler provided µ ≤ 1, and it is also irreducible provided µ < 1. Since there is only one interior equilibrium all orbits in the interior will converge to this equilibrium -cf. [START_REF] Hirsch | Stability and convergence in strongly monotone dynamical systems[END_REF]Theorem 10.3]. Therefore, the endemic equilibrium is a globally asymptotically stable equilibrium for System (18) even if condition (4) is not satisfied and hence the Lotka-Volterra function type (2) cannot be a Lyapunov function.

µ

1. Let us write µ = -1 , with 0 < 1. Then the AMG system can be written as:

         Ẋ = X s -X -(X s -1)M X, Ṁ = ν rY r -1 - M X r -1 -M , Ẏ = M X -Y. (19)
Since Y = M X is a globally stable equilibria of the last equation, we can use Tikhonov's Theorem [START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF][START_REF] Verhulst | Singular perturbation methods for slow-fast dynamics[END_REF] to conclude that solutions of System [START_REF] Li | Global stability for the seir model in epidemiology[END_REF] are close to solutions of the following reduced system:

(20) Ẋ = X s -X -(X s -1)M X, Ṁ = νM (X -1) .
In view of the fact that (1, 1) is a globally asymptotically stable equilibrium for System [START_REF] Li | A geometric approach to global-stability problems[END_REF], one expects that the endemic equilibrium is also globally asymptotically stable in this regime. This is the contents of the next result.

Proposition 4.1. For sufficiently small and positive values of the interior equilibrium (1, 1, 1) = 1 * of System (19) is globally asymptotically stable.

Proof. It is easy to check that there exits a compact absorbing K for System [START_REF] Li | Global stability for the seir model in epidemiology[END_REF]. In addition, since this system is also uniformly persistent, we can choose K with a positive distance to the boundary of the positive orthant. Let D and G be the orthogonal projections of K onto (X, M ) ≥ 0 and Y ≥ 0, respectively. Moreover, this set K can also be chosen such that D is a compact absorbing set for System [START_REF] Li | A geometric approach to global-stability problems[END_REF]. The reduced System (20) has (1, 1) as its unique interior equilibrium -which is always locally stable -provided X s > 1. Its global stability follows by noticing that ϕ(X, M ) = 1 M is a Dulac function for System [START_REF] Li | A geometric approach to global-stability problems[END_REF] in the positive quadrant. Notice also that as a differential equation in the fast time τ , Y τ = M X -Y is globally attractive on R + , provided M X > 0a condition that is always satisfied over D.

Therefore, we have that all hypothesis of Tikhonov's theorem are verified -cf. Finally, we also observe that since the solutions of System [START_REF] Li | Global stability for the seir model in epidemiology[END_REF] never leave D or G, the approximations provided by Tikhonov's theorem are valid globally in time -see the remark in [38, pg. 750]. More precisely, let (X , M , Y ) be the solution to [START_REF] Li | Global stability for the seir model in epidemiology[END_REF] with an interior initial condition (X 0 , M 0 , Y 0 ) and let (X 0 , M 0 ) be the solution to [START_REF] Li | A geometric approach to global-stability problems[END_REF] with initial condition (X 0 , M 0 ) -set Y 0 = M 0 X 0 . Then, if we write • ∞,τ to denote the uniform norm on [τ, ∞) we have, for sufficiently small , that

Z -Z 0 ∞,0 ≤ C 1 , with Z = X, M and Y -Y 0 ∞,τ ≤ C 2 , for some C 1 , C 2 , τ > 0 that can be taken independently of .
The remainder of the proof is split in two subcases that depend on the value of the parameter g. For 0 < g < 4, there is a neighbourhood U of 1 * such that VEE ≤ 0, vanishing only on the intersection of the critical line with U -recall the discussion in Section 3, from which follows that the size of U depends only on the quantity 4-g and hence that it can be taken independently of .

In particular, let δ = dist(1 * , ∂U ) > 0 and notice that for τ > τ , Z = X, Y, M and < δ 2 , we have that

Z -1 * ∞,τ ≤ Z -Z 0 ∞,τ + Z 0 -1 * ∞,τ ≤ C + Z 0 -1 * ∞,τ < δ 2 for large τ < δ.
Thus, global stability of 1 * follows from an application of LaSalle's principle. For g ≥ 4, further work is needed. In this case, the standard estimate for the size of the local basin of attraction of an equilibrium that is locally asymptotically stable scales withcf. [START_REF] Teschl | Ordinary differential equations and dynamical systems[END_REF]Theorem 3.26].

In order to ensure that the solution of System [START_REF] Li | Global stability for the seir model in epidemiology[END_REF] is trapped in this basin of attraction, we need to proceed to the next order in the expansion -see for instance [27, pages 46-51]. More precisely, let us write (X , Y , M ) = (X 0 , Y 0 , M 0 ) + (X 1 , Y 1 , M 1 ) + o( ).

In this case, inserting the above expansion into System [START_REF] Li | Global stability for the seir model in epidemiology[END_REF] and collecting the terms that are first order in we get, from the third equation of System [START_REF] Li | Global stability for the seir model in epidemiology[END_REF], an equation for Y 1 that read as follows:

Y 1 = -Ẏ 0 + X 0 M 1 + X 1 M 0 , and, from the first and second equations, we obtain d dt

X 1 M 1 = A(t) X 1 M 1 + b(t), X 1 (0) M 1 (0) = 0, with A(t) = M 0 (1 -X s ) -1 X 0 (1 -X s ) νM 0 ν(1 -X 0 ) and b(t) =   0 - rν r -1 Ẏ 0   .
The error estimate then becomes

Z -Z 0 -Z 1 ∞,0 ≤ C 3 2 , with Z = X, M and Y -Y 0 -Y 1 ∞,τ ≤ C 4 2 ,
for some C 3 , C 4 , τ > 0 that can be taken independent of -cf. the analysis on [27, pages 56-58]. Let

A ∞ = lim t→∞ A(t) = -X s 1 -X s ν 0 ,
and notice also that lim t→∞ b(t) = 0. Thus, for t sufficiently large we have that

A(t) = A ∞ + b(t) with b(t) small. Π A(t) (t, 0) = Π A(t) ( t, 0)Π A∞+b(t) (t, t) ≤ C Π A∞+b(t) (t, t) ≤ Ce -α(t-t) ,
for t > t and for some α, C > 0. Therefore, let U now denote the basin of attraction of 1 * and let δ = dist(1 * , ∂U ) > 0, with δ < C , and notice that for τ > τ , Z = X, Y, M we have

Z -1 * ∞,τ ≤ Z -Z 0 -Z 1 ∞,τ + Z 0 -1 * ∞,τ + Z 1 ∞,τ ≤ C 2 + Z 0 -1 * ∞,τ + Z 1 ∞,τ < 2 for large τ < C .
Thus, for t > t, the solution is trapped inside U and, hence, converge to 1 * .

Remark 4.1. It is also worthwhile to point out that the regime µ 1 is indeed relevant to malaria. The lifespan of a healthy blood cell is typically 40 days , while for an infected one is about one day -cf. [START_REF] Hetzel | The within-host cellular dynamics of bloodstage malaria: theoretical and experimental studies[END_REF]. Since µ = µ y µ x , values of 40 or more are typical.

ν 1.

A similar argument works in this case as well. The reduced system is now given by ( 21)

Ẋ = X s -X -(X s -1) rY X X + r -1 , Ẏ = µY rX X + r -1 -1 , where M = rY X + r -1 .
Notice also that ϕ(Y ) = 1 Y is a Dulac function for system [START_REF] Lin | Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations[END_REF]. See Figure 4 for a an example of the dynamics in this regime. Remark 4.2. This also a relevant regime for Malaria. Indeed, according to [START_REF] Hetzel | The within-host cellular dynamics of bloodstage malaria: theoretical and experimental studies[END_REF], for malaria in men we have ν 1920 4.5. An example with µ > 1 but with µ = O (1). Consider System (5) with the following parameters:

µ = ν = 6 5 , r = 3 2 and X s = 5.
Using a geometrical approach, we can show that the equilibrium is also globally asymptotically stable with these parameters.This can be done by carefully following a typical orbit and then showing that it either must converges to the endemic equilibrium or ending up in a faith set F. The choice of such a set carries some freedom, which can be explored to provide a 'good geometry', from which convergence to the endemic equilibrium follows. Further details will be provided elsewhere [START_REF] Iggidr | A geometrical approach to explore the stability of the AMG model[END_REF].

Discussion

Since their most recently rediscovery by [START_REF] Korobeinikov | A lyapunov function and global properties for sir and seir epidemiological models with nonlinear incidence[END_REF][START_REF] Lin | Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations[END_REF], the Volterra-type Lyapunov functions have been a workhorse in establishing global stability results in a variety of models in mathematical biology. This successful track record brought them a distinguished status as tools to prove global stability -cf. [START_REF] Fall | Epidemiological models and Lyapunov functions[END_REF] and [START_REF] Shuai | Global stability of infectious disease models using lyapunov functions[END_REF].

In this work, we revisited the study of the global dynamics of the AMG system -a well-known in-vivo model to Malaria that is also equivalent to a within-host HIV model. We then showed the following results: (i) there exists a unique Volterra function (up to scaling and an additive constant) that can be a candidate Lyapunov function; (ii) this function cannot be a Lyapunov function for the range of parameters that is biologically relevant. As a matter of fact, we present a necessary and sufficient condition for this Volterra candidate function be a Lyapunov function for the AMG system.

It is worthwhile to notice that, despite its apparent simplicity, a complete characterisation of the global dynamics of the AMG system has not been rigorously established yet. Indeed, it is believed that AMG system has the Sharp Threshold Property [START_REF] Shuai | Global stability of infectious disease models using lyapunov functions[END_REF], but so far a proof of this fact has been elusive -the proof that the endemic equilibrium is globally asymptotically stable has only been provided in certain cases. There are, nevertheless, a number of claims of complete proofs which are either incorrect or which are equivalent to conditions that cannot be verified.

In addition, we have enlarged the range of parameters where global stability of the endemic equilibrium holds by using a set of different approaches: (i) a change of state variables that turn the system in either (i-a) a strongly monotone system or (i-b) a triangular system that is asymptotically 2-D; (ii) exploit the existence of a large parameter and Tikhonov's theorem (together with some extensions) to obtain a reduced 2-D system that is globally stableand that this global stability implies the global stability of the full system. This extended parameter range include regimes that are relevant to Malaria. Furthermore, we have also provided an example of a set of parameter values for which we can also show global stability of the endemic equilibrium -this uses an alternative geometrical technique that will be discussed elsewhere [START_REF] Iggidr | A geometrical approach to explore the stability of the AMG model[END_REF].

Finally, we hope that these results will raise caution among mathematical modellers that, while the Lotka-Volterra function is a formidable tool, it also has its limitations. While this might sound obvious, we are not aware of any example where is rigorously shown that the Lotka-Volterra is not a Lyapunov function for all the relevant parameters. This failure shows that further tools for studying global stability are already needed to fully characterise biologically relevant systems as the AMG model.

4. 1 .

 1 The case µ = 1. Let N = αX + Y and S = αX + 2Y + α M , where α = µ X s -1 and

4. 2 .

 2 The case µ < 1. Instead of use S, let Z = Y + α M -so that S = N + Z. In the N ,Z,Y variables we have[START_REF] Kuznetsov | Elements of Applied Bifurcation Theory[END_REF]

  [38, Theorem 1.1]. Furthermore, the equation Y τ = M X -Y trivially satisfy the additional stability assumptions needed for the improved asymptotic results by O'Malley and Vasil'eva -cf. [38, Theorem 2.1].

  (a) g = 1.5 (b) g = 2 (c) g = 3.5 (d) g = 4 (e) g = 6 (f) g = 10

Figure 1 .

 1 Figure 1. Examples of p(X, T ) = 0 curves.

Figure 2 .

 2 Figure2. Parameter range where global stability of endemic equilibrium is known and approach used -Γ = X s 4(r -1/2) 2 . The parameter ν does not seem to play a fundamental role in the stability features, except by allowing a twodimensional reduction when it is large.

  (a) X-M projection (b) Y and M X (c) full picture

Figure 3 .

 3 Figure 3. Full and reduced solutions. Parameters: X s = 10, r = 2, ν = 3 and µ = 5000.

  (a) X-Y projection (b) Y and rY X + r -1 (c) full picture

Figure 4 .

 4 Figure 4. Full and reduced solutions. Parameters: X s = 10, r = 2, ν = 1500 and µ = 50.

reviewer for their careful reading and useful remarks, which allowed significant improvement of the manuscript. ‡ Funding MOS was partially supported by CNPq grant # 310293/2018-9 and by CAPES/BR -Finance code 001. AI acknowledges the financial support of the program STIC AmSud (project 20-STIC-05) and of Inria through the Associate Team MOCOVEC. This work was completed when AI was visiting UFF within the framework of CAPES/PRINT -project grant # 88881.310201/2018-01.

4.6.

A conjecture on global stability. In view of the global stability of the endemic equilibrium, 1 * , for both 0 < µ ≤ 1 and µ 1, together with the non dependence of R 0 on µ suggests that µ should be the relevant parameter for stability and, hence, that a bifurcation analysis with respect to µ might complete the global stability picture.

We begin by gathering some previous results in a convenient way:

Lemma 4.1. Assume that X s > 1 (which implies r > 1 and R 0 > 1). Then the following holds: (i) the AMG system is uniformly persistent; (ii)1 * is locally asymptotically stable; (iii) the DFE equilibrium is a saddle with an unstable manifold of dimension one. (iv) the AMG system is competitive. In particular, no homoclinic orbit may exist.

The following result is a recast of a result in [START_REF] Li | Global stability for the seir model in epidemiology[END_REF] and we provide a sketch of the proof for the convenience of the reader: Lemma 4.2. Assume X s > 1. If 1 * is not globally asymptotically stable, then there exists an unstable periodic orbit and this orbit lies on the boundary of basin of attraction of 1 * .

Proof. Let U be the basin of attraction At µ * , we must have that Σ = ∂U ∩ int(K) is nonempty -recall that K is a compact absorbing set associated to the AMG system. Hence Σ is invariant and must contain a periodic orbit γ (since it does not contain any equilibrium point). Notice also that this orbit is also in the alpha limit set of 1 * and, hence, cannot be stable. Lemma 4.3. Assume that X s > 1 and the endemic equilibrium is globally asymptotically stable. Then the AMG system is a Kupka-Smale system-and indeed a Morse-Smale system, since it is competitive.

Proof. (i) the equilibria is always hyperbolic; (ii) since the endemic equilibrium is globally asymptotically stable there are no periodic orbits; (iii) the unstable manifold of the DFE intersects the stable manifold of the endemic equilibrium transversally.

In what follows, we recall that K is an absorbing set for the AMG System and we will write U to denote the basin of attraction of 1 * . Proposition 4.2. Let µ * be the largest value such that 1 * is globally asymptotically stable on (0, µ * ). Then, generically, a saddle-node bifurcation of periodic orbits occurs at µ * .

Proof. From Lemma 4.2, there exists a periodic unstable orbit γ ∈ ∂U .

Let ζ 1,2 be the Floquet multipliers of γ, with |ζ

Since γ is a non structurally stable orbit (it does not exist for µ < µ * ) it is non-hyperbolic. Thus |ζ 2 | = 1, and hence ζ 2 = 1 -cf. [START_REF] Kuznetsov | Elements of Applied Bifurcation Theory[END_REF]. Therefore at µ = µ * a saddle-node bifurcation of periodic orbits must occur generically [START_REF] Kuznetsov | Elements of Applied Bifurcation Theory[END_REF]. Notice that this argument precludes, at least generically, period doubling and torus bifurcations. Remark 4.3. For µ > µ * , but with µ -µ * small, then γ is hyperbolic, and hence a saddle orbit, and another periodic orbit, stable, will also exists in this regime. Moreover, since the AMG system is analytic and relative prime, only a finite number of bifurcations might happen at µ = µ * -cf. Wintner's principle of natural termination. Notice also that there exists µ * * with 1 < µ * <µ * * such that the families of bifurcations born at µ = µ * will terminate at µ ≤ µ * * As a matter of fact, numerical simulations with µ, ν > 1 suggest that, as t → ∞, the solution approaches either the surface Y = M X or the surface M = rY X + r -1 -and, as we saw above, on these surfaces the system is globally asymptotically stable. , depending on which is the largest one. This implies that the AMG system satisfies the Sharp Threshold property for all µ > 0.


Andrea Maffia
email: andrea.maffia@unipv.it

Liliana Silva
email: liliana.silva@unime.it

On the use of boardgames to develop young children's number sense

Keywords: Boardgames, number sense, literature reviews

With the aim of nourishing the discussion that has taken place during CERMEs in TWG13, we present a review of results presented in previous CERMEs about the possible roles of boardgames in developing young children's number sense. Several contributions in CERMEs have shown that difficulties related to teacher education and curriculum design emerge when analyzing the boardgames that are proposed and their management in classrooms. However, the discussion about the contribution of boardgames to preverbal number sense is still underdeveloped when compared to literature from research journals. Also, the potential of boardgames in providing challenging mathematical tasks seems still unresearched. Hence, possible future paths of research for the use of boardgames in early years mathematics emerge.

Introduction

Play can be considered as the main learning experience for young children in general, and for early years mathematics in particular [START_REF] Schuler | Playing and learning in early mathematics education-modelling a complex relationship[END_REF]. Among the many possible playful activities in which children may engage, we can roughly distinguish between free play and guided play, the latter consisting of adults structuring of the play environment but leaving control to the children within the environment [START_REF] Weisberg | Guided play: where curricular goals meet a playful pedagogy[END_REF]. In this contribution we will focus on guided playful activities, to which we will refer as games. There is a large body of research about the use of games in learning (game-based learning), but not all kinds of games have been studied with the same attention. While literature about videogames is getting larger and larger (e.g. [START_REF] Yong | Let us rethink how to teach mathematics using gaming principles[END_REF], boardgames were rarely researched [START_REF] Ramani | Promoting broad and stable improvements in low-income children's numerical knowledge through playing number boardgames[END_REF]. We consider as boardgames all those games that are played on a printed surface by one or more people usually sitting around a table (Parlett, 2018); they might include the use of cards or dice. As testified by the available literature, boardgames can play an important role in the teaching and learning of mathematics (see next sections). Such relevance might be even higher at early grades, as stated in some contributions from previous CERMEs (e.g. [START_REF] Schuler | How can games contribute to early mathematics education? A video-based study[END_REF][START_REF] Schuler | Playing and learning in early mathematics education-modelling a complex relationship[END_REF][START_REF] Tubach | Children focusing on differences between numbers in the context of a playing environment[END_REF].

The aim of this paper is to nourish the discussion that has taken place during CERMEs in TWG13 by summarizing results presented in past proceedings. Such review of the literature is intended to suggest further limits and opportunities in using boardgames in early mathematics education. Our discussion is limited to number sense; the next session is devoted to explaining what we mean by that.

Number sense

The construct of 'number sense' is present since many years in the discourse of researchers in mathematics education in general, and in the ERME community in particular [START_REF] Rezat | Number sense in teaching and learning arithmetic[END_REF]. Several definitions have been proposed, mostly in the form of a list of abilities that a child should develop in order to show number sense. Such lists are often not equivalent. In this contribution we adopt the classification provided by Andrews and Sayers (e.g. Andrews & Sayers, 2015) who distinguish three categories of number sense.

Preverbal number sense includes the innate abilities necessary for quantitative understanding. Research literature (within and outside mathematics education) shows that very young children can discriminate small quantities (e.g. subitizing) and arrange them as a linear representation, called 'mental number line ' (e.g. Deahene, 2001).

Applied number sense refers to competences that prepare the learners for the adult life. Applied number sense should enable a person to look at a problem holistically before confronting details, look for relationships among numbers and operations and […] consider the context in which a question is posed; choose or invent a method that takes advantage of his or her own understanding of the relationships between numbers or between numbers and operations and […] seek the most efficient representation for the given task; use benchmarks to judge number magnitude; and recognize unreasonable results for calculations in the normal process of reflecting on answers [START_REF] Reys | Promoting number sense in the middle grades[END_REF], p. 115, as quoted in Andrews & Sayers, 2015).

Finally, there is a set of intermediate abilities that are developed thanks to instruction, usually between the end of preschool and the beginning of primary school; then they are particularly relevant for early years mathematics. This set of eight abilities, called foundational number sense, is listed in Table 1.

The distinction between preverbal, foundational, and applied number sense will serve as a framework for organizing our presentation of the reviewed literature. The next sections are devoted to describing if and how each of these three types of number sense was considered in past contributions to TWG13 about boardgames. Furthermore, we will list limits evidenced by researchers.

Selection of papers

The papers selected for the presented review are taken from proceedings of the last six editions of CERME. A mapping review of all the abstracts of the paper presented in the working group about 'Early Years Mathematics' (TWG14 in CERME6, when it was founded; TWG13 in the following edition) served for a first phase of selection. Among the 124 contributions, we selected those explicitly referring to numerical abilities and to a playful/game context. We then realized a second phase of selection by reading the full text; only papers specifically including boardgames were left. We ended up with the four papers listed in Table 2.

The fact that a low number of contributions were selected sustains the claim that research about boardgames is still underdeveloped. Surely, TWG13 is not the only TWG interested in boardgames and number sense (e.g. [START_REF] Sensevy | A model for examining Teacher's didactic action in mathematics, the case of the game "race to 20[END_REF], however as we will discuss later, we believe that this topic is particularly relevant for TWG13. We can notice that papers about these topics appeared in this working group during several editions of CERME; we may then affirm that a discussion has started, and there is still room to widen it. The four selected papers have been categorized referring to the framework introduced above. Results are presented in the following sections.

Boardgames and preverbal number sense

Research on preverbal number sense usually focuses on children aged 0-6 and then this stream of research should be considered as part of research in early years mathematics education. Indeed, TWG13 has collected contributions about subitizing (e.g., [START_REF] Schöner | Two, three and two more equals seven" Preschoolers' perception and use of structures in sets[END_REF], but none of them included the use of boardgames. One of the selected papers refers to subitizing [START_REF] Schuler | Playing and learning in early mathematics education-modelling a complex relationship[END_REF], but it is not the main focus of research. Surely this fact does not depend on the impossibility of using boardgames to assess or develop preverbal numbers sense. For instance, we may notice that many traditional boardgames (like Shut the Box) include the use of dice. Traditionally, cubic dice show quantities through dots arranged in a canonic way. Research shows how subitizing gets easier when children recognize patterns in the arrangement of dots; furthermore, developmental dyscalculia can cause a deficit in the estimation of canonically arranged dots [START_REF] Ashkenazi | Do subitizing deficits in developmental dyscalculia involve pattern recognition weakness?[END_REF]. This fact suggests that playing boardgames can help in recognizing specific difficulties that might depend on inexperience with canonical arrangement of dots or on learning disabilities. (1) Number recognition Recognition of number symbols; their vocabulary and meaning. Ability to identify a particular number symbol from a collection and name a number when shown.

(2) Systematic counting Counting systematically to twenty and back or count upwards and backwards from arbitrary starting points; knowing each number's position in the sequence of all numbers.

(3) Awareness of the relationship between number and quantity Not only understanding the one-to-one correspondence between a number's name and the quantity it represents, but also that the last number in a count represents the total number of objects. Another clear example is given by research conducted on the development of the mental number line using linear boardgames (boardgames with linearly arranged, consecutively numbered, equal-size spaces, e.g. Chutes and Ladders, Game of the Goose) -we are here referring to research that was not presented in CERME proceedings, but in research journals. According to [START_REF] Siegler | Development of numerical estimation in young children[END_REF] and to [START_REF] Ramani | Promoting broad and stable improvements in low-income children's numerical knowledge through playing number boardgames[END_REF]:

In such games, the greater the number in a square, the greater (a) the distance the child has moved the token, (b) the number of discrete moves the child makes, (c) the number of number names the child has spoken, (d) the number of number names the child has heard, and (e) the amount of time since the game began. The linear relations between numerical magnitudes and these visuospatial, kinesthetic, auditory, and temporal cues provide a broadly based, multimodal foundation for a linear representation of numerical magnitudes. (Ramani & Siegler, 2008, pp. 376-377) Research shows that playing these boardgames strengthen preschoolers' number line estimation, magnitude comparison, numeral recognition, and counting skills [START_REF] Siegler | Development of numerical estimation in young children[END_REF][START_REF] Ramani | Promoting broad and stable improvements in low-income children's numerical knowledge through playing number boardgames[END_REF]. Also, evidence shows that playing boardgames correlate positively with numerical knowledge, while this is not the case for videogames and card-games. Children from middle-income backgrounds reported playing more boardgames, and fewer videogames, than age peers from lowincome backgrounds. This is considered as one of the possible explanations for differences in the development of preverbal number sense in relation to socio-economic status [START_REF] Ramani | Promoting broad and stable improvements in low-income children's numerical knowledge through playing number boardgames[END_REF]. [START_REF] Whyte | Number games, magnitude representation, and basic number skills in preschoolers[END_REF] have compared linear boardgames with card-games and found that card-games can improve some aspects of children's number sense, but not numerical estimation (evaluated as positioning on a number line). Playing with linear boardgames helps children to shift from a logarithmic to a linear representation of numerical magnitudes [START_REF] Whyte | Number games, magnitude representation, and basic number skills in preschoolers[END_REF].

Boardgames and foundational number sense

All the contribution selected for our review focused on foundational number sense. The quality of the game and the role of the educators (parents or teachers) appear to be central for all the authors.

Foundational number sense needs explicit teaching (per definition) and thus the choice of a boardgame should be guided by the aim of a teaching intervention. Following Brousseau's Theory of didactical situations, an appropriate game is selected when it allows "bringing together a 'milieu' and a 'player', with this game being such that a given piece of knowledge will appear as the means of producing winning strategies" (Brousseau, 1998, p.57). The equilibrium between the game and the mathematical content seems hard to achieve, as has been shown by contributions to previous CERMEs. On one side, the game should be mathematically productive, meaning that the game materials should be reinterpreted as representations of mathematical relationships [START_REF] Tubach | Children focusing on differences between numbers in the context of a playing environment[END_REF].

On the other side, it is important that the game is an actual game to children for exploiting the game's idea and affordance [START_REF] Schuler | How can games contribute to early mathematics education? A video-based study[END_REF]. However, there is also the risk that a game remains just a game. As reported by [START_REF] Schuler | Playing and learning in early mathematics education-modelling a complex relationship[END_REF]Schuler ( , p. 1920) while referring to a boardgame about counting and comparison of numbers:

Mathematical potential develops through the educator's comments on the game's course, through questions that stimulate explanations, reflections on actions and thoughts, and reasoning. She has to communicate individually challenging rules through stimuli, comments, questions and requests what requires a sensitivity for possibilities and variations in the games course. [START_REF] Dorier | Didactical analysis of a dice game[END_REF] refers about teachers selecting games based on the pleasure they are supposed to give to students, while the mathematical content remains secondary. They analyze a game called 'Turn the Dice' proposed for first grade in the official curricular material of the Frenchspeaking Switzerland. By their analysis, to play correctly, students should know how to make sums correctly. However, if they do not, they may play anyway, because nothing in the managing of the game is organized to provide any feedback. In their words "nothing is organized didactically for them to learn sums, they have to know, but they can make errors without being corrected, except if the other player knows better or the teacher is there to correct" [START_REF] Dorier | Didactical analysis of a dice game[END_REF], p. 2580). They also observed a teacher conducting the game in one of her classes and found how she had probably underestimated the difficulty of the game.

While research elsewhere as shown how boardgames may provide a good opportunity for the development of foundational number sense [START_REF] Peters | Playing games and learning mathematics: The results of two intervention studies[END_REF][START_REF] Stebler | Play-based mathematics in kindergarten. A video analysis of children's mathematical behaviour while playing a boardgame in small groups[END_REF], the discussion going on in TWG13 has pointed out possible limits for the exploitation of such potential.

Boardgames and applied number sense

Potentialities of boardgames for developing applied number sense are as many as the different representations of numbers, calculation algorithms, and so on. In the context of TWG13, the interest could be in understanding how boardgames may help, in the context of early years mathematics, to foster the development from foundational to applied number sense. In one of the selected papers, [START_REF] Dorier | Didactical analysis of a dice game[END_REF] note that even simple games can hide interesting opportunities for introducing more complex mathematics. For instance, they show how, by slightly changing the rules of the game, 'Turn the Dice' shows strong similarities with the famous game 'Race to 20' which offers interesting opportunities to introduce the Euclidean division [START_REF] Sensevy | A model for examining Teacher's didactic action in mathematics, the case of the game "race to 20[END_REF].

This example helps us in introducing an opportunity provided by boardgames that has been studied scarcely. In their paper from the Journal für Mathematik-Didaktik, after analyzing children playing the game Shut the Box, Stebler and colleagues conclude that:

This first exploratory analysis leads us to the hypothesis that boardgames like Shut the Box can provide a high-quality teaching and learning arrangement, offering cognitively activating and challenging learning tasks, adaptive for different levels of mathematical competencies and allowing for diverse strategies, embedded in a collaborative setting (2013, p. 172) Such hypothesis needs confirmation, but the possibility of using boardgames as context for mathematical activity that is adaptive for different levels of competencies is proposed by other authors in research literature (e.g. [START_REF] Vogt | Learning through playpedagogy and learning outcomes in early childhood mathematics[END_REF]. Some authors notice that the challenges that are posed by a boardgame are particularly effective for children with higher mathematical competences, while more traditional training program could lead to better results for low achievers [START_REF] Vogt | Learning through playpedagogy and learning outcomes in early childhood mathematics[END_REF]; other authors suggest using boardgames even in the case of disabled children since they prove to be successful with students of different ability levels [START_REF] Mcconkey | Games for learning to count[END_REF].

Conclusion

General education literature converges on attesting that game-based education can foster learning providing students with a high motivational context, and this is also confirmed in the case of mathematics [START_REF] Yong | Let us rethink how to teach mathematics using gaming principles[END_REF]. Research about motivation and games is widespread in journals and conferences, but we can recognize that the reviewed research focuses more on the cognitive and relational aspects that are specific to the development of number sense using boardgames. The focus on the different didactical variables, including the teacher's role, appears peculiar to TWG13.

Drawing on literature from research journals, we have noticed that linear boardgames and playing with dice can help in fostering preverbal number sense (specifically, the mental number line and the subitizing ability) and in assessing deficits, however research about boardgames and preverbal number sense is still missing in CERME. Contributions in CERMEs have shown that difficulties related to teacher education and curriculum design emerge when analyzing the boardgames that are proposed in mathematics classes. However, we believe that these difficulties do not constitute good enough reasons to avoid the use of boardgames in preschool or primary school, since there is evidence that these games can provide the context for challenging tasks. [START_REF] Downton | Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections[END_REF] have documented (in the context of word problems) how challenging tasks may prompt the use of more sophisticated calculation strategies and then flexibility. This could be particularly true in the case of strategic games. Further research is needed to prove this kind of conjectures. If any, positive results may serve as basis for a larger implementation of boardgames in mathematics classes.

Research literature testifies that teachers participating in experimental interventions involving boardgames are particularly enthusiast [START_REF] Vogt | Learning through playpedagogy and learning outcomes in early childhood mathematics[END_REF], and such involvement is considered one of the possible causes of good results of these experiments [START_REF] Ramani | Promoting broad and stable improvements in low-income children's numerical knowledge through playing number boardgames[END_REF]. Enjoyment may lead teachers to propose tasks that are more challenging than those they are used to and then prompts an explorative approach that is often prevented by teachers' insecurities [START_REF] Peters | Playing games and learning mathematics: The results of two intervention studies[END_REF].

However, as noted above, teachers may misjudge the mathematics involved in a boardgame [START_REF] Dorier | Didactical analysis of a dice game[END_REF]. The design of teacher professional development programs appears as needed and may constitute an interesting context of research [START_REF] Maffia | Primary teachers' professional development about mathematics assessment using tabletop games[END_REF].

Concluding, we can claim that boardgames show several potentialities in relation to the development of young children's number sense, however in TWG13's discussion there is still room for more research aimed at: (1) understanding the features of boardgames for developing and assess preverbal and foundational number sense; (2) developing teachers' education to help them in analyzing games and exploit their potential.

  (4) Quantity discrimination Understanding magnitude and comparing of magnitudes. Use of language like bigger than or smaller than.(5) Understanding of different representations of number Understanding that numbers can be represented differently, including the number line, different partitions, various manipulatives and fingers.(6) Estimation Estimation, whether the size of a set or an object. Moving between representations of number; for example, placing a number on an empty number line. sequences and identification of a missing number in simple number sequences.

Table 1 : Components of foundational number sense (Andrews & Sayers, 2015)

 1

Table 2 : Selected papers from past CERME proceedings

 2

	Author(s)	Title


Hanna Palmér
email: hanna.palmer@lnu.se

Camilla Björklund
email: camilla.bjorklund@ped.gu.se

Toddlers' ways of experiencing the meaning embedded in the question "How many years are you?"

Keywords: early mathematics, finger pattern, numbers, representations, toddlers

This paper focuses on toddlers ' (1-to 3-year-olds) ways of responding to the question "How many years are you?" and in what respect their responses may connect to a mathematical meaning. As a part of a longitudinal study on toddlers' numerical development, the question was posed to 15 toddlers five times at approximately six-month intervals. At the beginning of the study, no toddler responded to the question with a mathematical representation. The results show how the toddlers started to discern the question as one to be answered with a mathematical representation, either number words or a finger pattern. The results also show how some of the toddlers started to connect these two modes of representations. In the paper, we also explore the different ways the toddlers used finger patterns to answer the question.

Introduction

This paper focuses on toddlers' (1-to 3-year-olds) ways of responding to the question "How many years are you?" (direct translation from the Swedish expression "Hur många år är du?"). The starting point for this interest was a Swedish combined research-development project on toddlers' numerical development. Even though there is a growing consensus in research and practice that children's numerical competencies start to develop at a very early age, few tools have been developed to make their progress in learning visible and researchable. Thus, one part of the project was to develop a research tool to make toddlers' ways of experiencing numbers researchable [START_REF] Björklund | Designing a tool for exploring toddlers' numerical competencies in preschool[END_REF]. In this paper, we take a closer look at one of the questions used in that tool.

There are challenges in exploring young children's numerical competencies, since verbal utterances cannot be taken as the primary source for analysis of their understanding. Furthermore, it is crucial to establish joint attention between child and researcher in a conversation or task to maintain the focus on a common objective [START_REF] Siraj-Blatchford | A focus on pedagogy: Case studies of effective practice[END_REF]. When developing the research tool, we thus wanted to make it possible for the young children to relate to and reason about the content based on their previous experiences as well as enable them to express different ways of experiencing numbers. Based on methodological reviews on studies involving children, we knew that, especially in research with very young children, multiple-method activity approaches had proven successful [START_REF] Aubrey | That child needs a good listening to! Reviewing effective interview strategies[END_REF]. Thus, the decision was made to design a task-based interview that included a variety of tasks and manipulatives making it possible for the toddlers to express themselves in different ways. This task-based interview was to be held in a naturalistic setting, placing the children at the centre of focus, recognizing their social and cultural experiences and understanding. The context of the interview became the birthday of Kitty the cat as even toddlers have experienced birthdays. Also, the use of a toy cat was intended to engage the children's interest and promote joint attention.

As an introduction to the interview, the children got to see and, if they wanted to, hold Kitty the cat. The children were told that it was the cat's birthday. Then the question "How many years are you?" was posed to the children. The question had a twofold purpose: partly to arouse curiosity and interest by inviting the children to a joint conversation and partly to see how the children responded to this question, which they had most likely heard before. At one point, when we presented the study at a national research conference, a researcher in mathematics education asked whether the question "How many years are you?" is a mathematical question. Inspired by that question, we here analyse how toddlers responded to the question. Our approach is that it is not up to us to decide whether the question is mathematical or not; it depends on the toddlers' ways of experiencing the meaning embedded in the question. Whether or not the question "How many years are you?" is mathematical is thus an empirical question, which this paper will focus on by answering : How do toddlers respond to the question "How many years are you?" and in what respect do their answers connect to a mathematical meaning?

Four fundamental aspects of numbers

We cannot empirically investigate whether children experience the question as a mathematical question; we can only investigate their responses to the question and whether their responses include mathematical representations. Together with ordinality, cardinality and part-whole relations, representations are fundamental aspects of the complex construct of numbers that are known to be necessary for children to learn about in order to develop their numerical competence (Björklund et al., 2021;[START_REF] Baroody | Early number and operations: Whole numbers[END_REF]. The meaning of ordinality implies that every object or number word has an exclusive position in a sequence and relates to the others in the same sequence [START_REF] Fuson | Some further clarifications of numerical terminology using results from young children[END_REF]. "How many" questions mostly refer to the meaning of cardinality in numbers, that is, number words are used to describe a coherent set. This is often observed in counting acts where the last uttered number word includes all the counted items [START_REF] Gelman | The child's understanding of number[END_REF]. Knowing that number words represent a set or composed unit of items is, further, a requisite for understanding numbers' partwhole relations, which allows children to compare sets, add, subtract and in different ways operate with numbers to solve numerical problems [START_REF] Venkat | Architecture of mathematical structure[END_REF]. Thus, number words include several aspects that become significant for making use of them, and their meaning is also connected to the context in which the words are used (see [START_REF] Fuson | Relationships between counting and cardinality from age 2 to age 8[END_REF]. As numbers are abstract by nature they have to be represented in some way in all communicative situations, for example, by number words as exemplified above. [START_REF] Lesh | Applied mathematical problem solving[END_REF] emphasizes five modes of representation that are important in mathematics education, where the learning (for example, of numbers) is reflected in the ability to make connections between and within these modes of representation: Real-World Situation, Pictures, Verbal Symbols, Written Symbols and Manipulatives. But, even if toddlers respond to the "how many" questions, for example, with finger patterns or number words, it is not self-evident that the representation used mediates a mathematical meaning. The meaning of representations has to be learnt through communication with others whereby representations assigned to children's actions mediate mathematical meaning [START_REF] Van Oers | Emergent mathematical thinking in the context of play[END_REF].

Theoretical framework

The theoretical framework in this study is the variation theory of learning [START_REF] Marton | Necessary conditions of learning[END_REF] which brings to the fore the child's perspective, or more precisely, children's ways of experiencing a phenomenon (see also [START_REF] Sommer | Child perspectives and children's perspectives in theory and practice[END_REF]. In every situation, several aspects of a certain phenomenon can be discerned and the aspects that are discerned are decisive for how the phenomenon is experienced by the child. The theory directs attention towards children's learning and explains learning in terms of changes in the child's ways of experiencing a phenomenon. In the analysis, how children respond to the question "How many years are you?" and changes in these responses are interpreted as expressions of discerned and undiscerned aspects. As we in the study focus on the emergence of numerical awareness among toddlers, non-numerical meaning is just as important to observe.

Method

The project was conducted in close collaboration between researchers and three preschool teachers in Sweden over two years. The project, including development of tools and methods, has been approved by the Swedish ethical review authority (Dnr: 2019-01037). During the longitudinal study, the play-based interview was conducted five times at approximately six-month intervals. The interviews were all video-recorded, with the written consent of the toddlers' legal guardians. The interviews were conducted by the preschool teachers. This was important as it is not reasonable to expect that a researcher who does not know the toddlers would be able to provide the children with the best conditions to show their knowledge. As toddlers' expressions are very subtle, the interviewer needs to have extensive knowledge of the individual children's ways of expressing themselves.

Altogether, 110 video-recordings focusing on the question "How many years are you?" were transcribed. Also, how the children responded to the follow-up question, "Can you show with your fingers?", was noted. This follow-up question was asked to the children who did not use finger representation in relation to the first question. The results presented in this paper are based on analysis of 75 of these video-recordings from those children who were between 12 and 18 months when the project started and who took part in all five interviews. This was in total 15 children born between January and September 2018. The children's responses were first categorized based on correct or incorrect answers and then inductive based on the representations they used. Then, each inductive category of representation was analysed focusing on discerned and undiscerned aspects in line with variation theory principles [START_REF] Marton | Necessary conditions of learning[END_REF]. This way of analysing the data provides information on toddlers' ways of experiencing the phenomenon that in this study is the question "How many years are you?".

Results

First, the results from the inductive analysis on the representations used will be presented. Each category (C) of representation will be elaborated on separately, then followed by an elaboration of the categories related to the longitudinal perspective constituting the five interviews. Finally, the toddlers' use of finger patterns in all the interviews are focused on. Table 1 presents an overview of the 15 toddlers' responses to the questions "How many years are you?" (Q1) and "Can you show with your fingers?" (Q2) in the five interviews (Int. 1-5). In C1, the children gave no apparent mathematical response to Q1. However, many of them did interact with the interviewer, for example, by talking about Kitty the cat. Some children did use numbers and/or fingers as responses in other tasks but did not seem to discern any numerical aspects in this question. For example, one boy did not respond to this question, but afterwards when the interviewer showed him two fingers asking "Is this three?", he answered "No, it's two." Thus, this category includes both children who did not use mathematical representations at all in the interview and children who did not experience this specific question as one to be answered with a mathematical representation. The nature of the question here comes to the foreground since it is likely to be more challenging for the child to present a number word as an answer to a "how many" question (regardless of the items asked for) than to determine the quantity of a (re)presented number (for example shown finger pattern).

In C2a and C2b, the children responded by saying either an incorrect (2a) or a correct (2b) number word. Both responses indicate that these children experience this question as one to be answered with a number words. However, based on only this question it is not possible to determine whether the number words used by the children have a mathematical meaning, for example ordinal or cardinal meaning, or not.

In C3a and C3b, the children responded by showing either an incorrect (3a) or a correct (3b) finger pattern. Both responses indicate that these children experience this question as a question possible to be answered with a mathematical representation. What distinguishes this way of responding to the question from the verbal number words is that the question was posed in one mode of representation (verbal) but the children responded in another mode of representation. However, based on only this question it is not possible to determine whether the finger patterns used by the children have a mathematical meaning for them, it may be a learnt procedure.

In C4a and C4b, the children responded by saying a number word and showing a finger pattern at the same time. The number word and/or finger pattern may be incorrect (4a) or correct (4b). Both responses indicate that these children experience this question as a question possible to be answered with mathematical representations and that they have discerned a connection between modes of representation. However, as previously mentioned it is not possible to determine whether the number words and/or finger patterns used have a mathematical meaning for the children.

As mentioned, the second question (Q2) was only asked to the children who did not use fingers in response to the first question (thus C1 and C2ab). Some children gave no response (5c) while others responded by showing either an incorrect (5a) or a correct finger pattern (5b). Some children showed a finger pattern consistent with the number word used to answer the first question. Thus, they may have been connecting these two modes of representation. Some children who had not answered the first question now showed the correct finger pattern, which may be a representation with mathematical meaning but could also indicate that the question "Can you show with your fingers?" induced a procedure (showing a finger pattern) not grounded in any discerned mathematical meaning.

A longitudinal perspective

In the first interview, the age of youngest child was 12 months and the oldest was 18 months. As the intention in the project was to initiate the investigations before the toddlers had begun to use numbers, the toddlers' limited responses that may be connected to mathematics in interview 1 were expected.

In the second interview (age span 18-24 months), 13 children gave no mathematical response. Two children responded to the first question, one with an incorrect and one with a correct number word, "one", in combination with showing one finger. To Q2 (asked to the 14 children that did not use fingers in Q1) six children responded with finger patterns, three of them correctly. As mentioned, this representation may or may not be connected to a mathematical meaning.

In the third interview (age span 21-28 months), eight children gave no mathematical response. Seven children responded correctly to Q1, six used number words and one used a finger pattern. The use of number words and finger patterns indicates that these children had an expectation to answer with a (to the observer) mathematical representation. The children aged one answered "one", and the children aged two answered "two". Thus, they may not have known "how many" one or two is, but they experienced that these representations can be used to answer a "how many" question. To Q2 (asked to the 14 children that did not use fingers in Q1) six children responded with finger patterns, four of them correctly. As mentioned, this representation may or may not be connected to a mathematical meaning.

In the fourth interview (age span 27-34 months), two children gave no mathematical response. Six children responded with a number word on Q1, five of them correctly. Three children responded with finger patterns, two correctly. Four children responded with number words in combination with finger patterns, thus coordinating two representations. Three of them did this correctly. Similar to the above, these children may not have known "how many" one or two is regardless of representation, but they seemed to know that this was a way to answer this question. On Q2 (asked to the eight children that did not use fingers in Q1), six children used finger patterns, two of them correctly.

In the fifth interview (age span 33-40 months), three children gave no mathematical response. Two children responded to Q1 with finger patterns, one correctly and one incorrectly. 10 children responded to Q1 with number words in combination with finger patterns, thus coordinating two representations. Nine of them did this correctly. Similar to the above, these children may not have known "how many" one or two is regardless of representation, but they seemed to know that this was an answer to this question. On Q2 (asked to the three children that did not use fingers in Q1), two children used finger patterns, incorrect.

How do toddlers use finger patterns?

In the five interviews, three different ways to use finger patterns as a response to Q1 and Q2 emerged:

1) Respond by showing all fingers (three examples in Figure 1). This way of using finger patterns indicates that the children have discerned a representational aspect, but this is not yet connected to any numerical aspect. Nevertheless, the questions can be answered by showing finger patterns as representation, although the finger patterns used are not yet mathematical representations. 2) Respond by showing a finger pattern with a selected number of fingers, but not the correct number. Some children instantly selected a number of fingers, others worked to fold down a number of fingers with the help of the other hand. This way of using fingers indicates that they have discerned that different finger patterns mean different things and it is therefore necessary to be precise when creating the finger patterns. It is possible that these finger patterns are experienced as representations of a "unit". But because of the lack of connection to the number asked for, it is likely that the finger patterns are still undifferentiated in an exact cardinal sense.

3) Respond by showing a correct finger pattern (three examples in Figure 2). Some children used fingers on one hand and others used fingers from two hands, the last when showing two by using the index fingers close together. Similar to the above, this way of using fingers indicates their discerning that finger patterns have a specific meaning and may also be based on an experience of finger patterns as representing a "unit". Some sense of cardinality may here come through in that "two" is not merely an act of creating a pattern but an intended action of bringing one finger on each hand together to compose a set.

Discussion

The questions elaborated on in this paper are how do the toddlers respond to the question "How many years are you?" and in what respect do their answers connect to a mathematical meaning? In accordance with the variation theory of learning [START_REF] Marton | Necessary conditions of learning[END_REF], how children respond to the question is interpreted as expressions of discerned and undiscerned aspects. Someone might argue that the toddlers, over the course of the five interviews, learnt which number words or finger patterns to use. However, one delicate issue is that the correct answer to this specific question changed with each birthday. Thus, the children changed their answers between the interviews (correct or incorrect), which indicated that they had not learnt one specific way to answer this question by taking part in the interviews. The question is quite difficult form a mathematical perspective based on the lack of visible set to connect to the question "how many". The results show that the toddlers initially did not discern aspects of the question that brought them to respond with any mathematical representations. One explanation may be that the toddlers had not yet discerned mathematical representations and therefore these were not part of their possible responses to the question. However, this is not true for all the toddlers as there were toddlers who did not respond to this question with a mathematical representation but used mathematical representations when responding to other questions in the same interview. Throughout the study the children on group level discern that the question can be answered by using what we interpret to be mathematical representations (number words and/or finger patterns). However, based on the analysis in this paper, we cannot know if the children do experience mathematical meaning of the representations that they use [START_REF] Van Oers | Emergent mathematical thinking in the context of play[END_REF]. To know the explicit meaning the representations have for the children, more situations in which they use representations must be analysed. Nevertheless, recent findings have shown that young children who make use of several representations, whether used in correct correspondence with a number of objects or not, are more attentive to learn cardinal meaning of numbers [START_REF] Gibson | Number gestures predict learning of number words[END_REF]. [START_REF] Lesh | Applied mathematical problem solving[END_REF] emphasizes five modes of representation, three of which are at play in this task, real-life situation provides context for the task while the toddlers respond by using number words and/or finger patterns. The children most often used one representation, then they started to use two representations in combination to a larger extent. However, the toddlers may not experience them as two different representations but as a whole, and they may not have known that the representations were mathematical; that definition is in the eyes of the observer.

Figure 1 :

 1 Figure 1: Different ways of showing all fingers

Figure 2 :

 2 Figure 2: Showing two with one or two hands

Table 1 : The children's responses to Q1 "How many years are you?" and Q2 "Can you show with your fingers?" (asked to children who did not use finger representation in answer to the first question)

 1

	Int. 1	Int. 2	Int. 3	Int. 4	Int. 5


Marja Van Den Heuvel-Panhuizen
email: m.vandenheuvel-panhuizen@nord.no

Iliada Elia
email: elia.iliada@ucy.ac.cy

Dag Oskar Madsen
email: dag.o.madsen@nord.no

Eleni Deliyianni
email: deliyianni@te.schools.ac.cy

Cathrine Bjerknes
email: cathrine.bjerknes@nord.no

Early childhood teachers' awareness of the characteristics of picture books for learning mathematics

Keywords: early mathematics, picture books, teachers' awareness

Research topic and theoretical framework

There is growing research which provides evidence that the use of picture books in the early years of schooling can contribute to the learning of mathematics (e.g., [START_REF] Van Den Heuvel-Panhuizen | Effects of reading picture books on kindergartners' mathematics performance[END_REF][START_REF] Van Den Heuvel-Panhuizen | The role of picture books in young children's mathematical learning[END_REF]. However, it is also known that different books can generate different amounts and different kinds of mathematical talk [START_REF] Anderson | Parents using picture books to support mathematical thinking[END_REF]. Therefore, it is essential to know what picture books can offer to elicit mathematical thinking of children. In particular, this knowledge is important for early childhood (EC) teachers when they use picture books to create opportunities for children to gain informal mathematical experiences. The study presented in this poster is a part of a research project which is carried out to acquire cognizance of how to enhance teachers' competence in using picture books in early childhood mathematics education.

The study builds on our previous research in which we developed and researched a framework of learning-supportive characteristics of picture books for learning of mathematics (Van den [START_REF] Van Den Heuvel-Panhuizen | Developing a framework for the evaluation of picturebooks that support kindergartners' learning of mathematics[END_REF]. This framework consists of two parts. Part A incorporates the mathematical content that is offered in a picture book. Besides the usual content domains this also refers to mathematical processes and attitudes, and mathematics-related themes. Part B describes how the mathematics is presented. It includes both the way of presentation and the quality of it.

The goal of the present study is to investigate whether the teachers are aware of the learningsupportive characteristics of picture books, specifically whether they are aware what mathematics can be found in picture books and the way the mathematics is presented, and whether the framework is of use for recognizing opportunities of picture books to support children's mathematical development. The latter is the focus of the poster.

Method

To investigate whether the framework helps to improve teachers' awareness of the opportunities picture books offer for early childhood mathematics education, a qualitative study was conducted in Cyprus and Norway with respectively 6 and 4 in-service EC teachers, who had to evaluate two picture books with and without the framework by filling in for each book two times an open questionnaire specially developed for each book. The picture books are regular trade books of high literary quality, not purposely written for teaching mathematics, but they tell appealing stories which form a meaningful context for engaging children in mathematical thinking. In both countries we used the same wordless picture books. In this way teachers have also more freedom to recognize learningsupportive characteristics and can develop their own ideas of using them. In the data analysis the focus was on finding noteworthy changes in teachers' answers between the two rounds.

Results and discussion

Overall, we found that in both rounds teachers' answers involved to a greater extent learningsupportive characteristics included in the supply of mathematical content (Part A) than characteristics referring to the presentation of mathematical content (Part B). Moreover, the answers often were short and without explaining their thinking. They rarely referred to mathematical processes or dispositions. Possibly they are not accustomed to using picture books or the framework did not appeal to them. Regarding the mathematics that was found, the framework helped the teachers to see different or more mathematics and further specify the identified mathematics from the first round. The framework also had a positive contribution to teachers' evaluation of how the mathematics is presented. However, in some cases the framework did not have added value. In the second round a few teachers just used the wording of the framework, while their answers earlier were more detailed and relevant.

Our study has shown that the framework has potential to make teachers aware of the learningsupportive characteristics of picture books for supporting children's learning of mathematics. Our next step will be to use our findings to set up a professional development for EC teachers.